From 6d9d85e08481ab281ecb316db6570d5e538689a9 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Mon, 1 Sep 2025 02:13:35 +0200 Subject: [PATCH 01/37] added completely uniformizable --- properties/P000221.md | 20 ++++++++++++++++++++ theorems/T000774.md | 9 +++++++++ theorems/T000775.md | 12 ++++++++++++ theorems/T000776.md | 15 +++++++++++++++ 4 files changed, 56 insertions(+) create mode 100644 properties/P000221.md create mode 100644 theorems/T000774.md create mode 100644 theorems/T000775.md create mode 100644 theorems/T000776.md diff --git a/properties/P000221.md b/properties/P000221.md new file mode 100644 index 0000000000..3f12b35316 --- /dev/null +++ b/properties/P000221.md @@ -0,0 +1,20 @@ +--- +uid: P000221 +name: Completely uniformizable +refs: + - wikipedia: Uniform_space + name: Uniform space on Wikipedia + - doi: 10.1007/978-1-4615-7819-2 + name: Rings of Continuous Functions (Gillman & Jerison) +--- + +There exists a complete uniformity on $X$. + +For definition of complete uniformity see . + +Compare with definition in 15.7 of {{doi:10.1007/978-1-4615-7819-2}} where uniform structure is defined using pseudometrics instead. + +---- +#### Meta-properties + +- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does. diff --git a/theorems/T000774.md b/theorems/T000774.md new file mode 100644 index 0000000000..84d37d06e5 --- /dev/null +++ b/theorems/T000774.md @@ -0,0 +1,9 @@ +--- +uid: T000774 +if: + P000221: true +then: + P000012: true +--- + +{P12} spaces are precisely the spaces admitting a uniformity. diff --git a/theorems/T000775.md b/theorems/T000775.md new file mode 100644 index 0000000000..01a7f47b1c --- /dev/null +++ b/theorems/T000775.md @@ -0,0 +1,12 @@ +--- +uid: T000775 +if: + P000162: true +then: + P000221: true +refs: + - doi: 10.1007/978-1-4615-7819-2 + name: Rings of Continuous Functions (Gillman & Jerison) +--- + +See corollary 15.14 of {{doi:10.1007/978-1-4615-7819-2}}. diff --git a/theorems/T000776.md b/theorems/T000776.md new file mode 100644 index 0000000000..2e5b06dd6b --- /dev/null +++ b/theorems/T000776.md @@ -0,0 +1,15 @@ +--- +uid: T000776 +if: + and: + - P000001: true + - P000164: true + - P000221: true +then: + P000162: true +refs: + - doi: 10.1007/978-1-4615-7819-2 + name: Rings of Continuous Functions (Gillman & Jerison) +--- + +A {P221} {P1} space is {P6}. Now apply theorem 15.20 of {{doi:10.1007/978-1-4615-7819-2}}. From 13b6476fe3086b0f3323615e00fcdd706df61bfd Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Tue, 2 Sep 2025 00:01:29 +0200 Subject: [PATCH 02/37] added --- properties/P000221.md | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/properties/P000221.md b/properties/P000221.md index 3f12b35316..9917977c02 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -1,14 +1,19 @@ --- uid: P000221 -name: Completely uniformizable +name: Dieudonné complete +aliases: + - Completely uniformizable + - Topologically complete refs: - wikipedia: Uniform_space name: Uniform space on Wikipedia + - wikipedia: Completely_uniformizable_space + name: Completely uniformizable space - doi: 10.1007/978-1-4615-7819-2 name: Rings of Continuous Functions (Gillman & Jerison) --- -There exists a complete uniformity on $X$. +The topology on the space $X$ is induced by some complete uniformity $\mathcal{U}$. For definition of complete uniformity see . From 6e49252a06cf4def5914f646922630a1ac80336d Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Tue, 2 Sep 2025 06:34:30 +0200 Subject: [PATCH 03/37] Kelley --- properties/P000221.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/properties/P000221.md b/properties/P000221.md index 9917977c02..7f22cd8772 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -11,6 +11,8 @@ refs: name: Completely uniformizable space - doi: 10.1007/978-1-4615-7819-2 name: Rings of Continuous Functions (Gillman & Jerison) + - mr: 370454 + name: General Topology (Kelley) --- The topology on the space $X$ is induced by some complete uniformity $\mathcal{U}$. @@ -19,6 +21,8 @@ For definition of complete uniformity see Date: Tue, 2 Sep 2025 07:55:53 +0200 Subject: [PATCH 04/37] added Engelking and equivalent definitions --- properties/P000221.md | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/properties/P000221.md b/properties/P000221.md index 7f22cd8772..4889733c13 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -13,11 +13,16 @@ refs: name: Rings of Continuous Functions (Gillman & Jerison) - mr: 370454 name: General Topology (Kelley) + - zb: "0684.54001" + name: General Topology (Engelking, 1989) --- -The topology on the space $X$ is induced by some complete uniformity $\mathcal{U}$. +The topology on the space $X$ is induced by a [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness) $\mathcal{U}$. -For definition of complete uniformity see . +If $X$ is $T_0$ this is equivalent to the following (see {{zb:0684.54001}} exercise 8.5.13a): + +2. $X$ is a closed subspace of a product of {P55} spaces +3. $X$ is a closed subspace of a product of {P53} spaces. Compare with definition in 15.7 of {{doi:10.1007/978-1-4615-7819-2}} where uniform structure is defined using pseudometrics instead. From dcf557bbb880ccf517f17a27376daea777d9268f Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Tue, 2 Sep 2025 08:01:47 +0200 Subject: [PATCH 05/37] added another justification to a theorem --- theorems/T000775.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/theorems/T000775.md b/theorems/T000775.md index 01a7f47b1c..6551d59d05 100644 --- a/theorems/T000775.md +++ b/theorems/T000775.md @@ -9,4 +9,5 @@ refs: name: Rings of Continuous Functions (Gillman & Jerison) --- -See corollary 15.14 of {{doi:10.1007/978-1-4615-7819-2}}. +See corollary 15.14 of {{doi:10.1007/978-1-4615-7819-2}} for complete uniformity on a {P162} space. +Alternatively, a {P162} space is a closed subspace of product of {S25} and {S25|P53}. From a351154e32fef5ec4fe7d76446f100e91a29e04e Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Tue, 2 Sep 2025 08:26:29 +0200 Subject: [PATCH 06/37] T386 change --- theorems/T000386.md | 10 ++-------- 1 file changed, 2 insertions(+), 8 deletions(-) diff --git a/theorems/T000386.md b/theorems/T000386.md index 2664032d12..6ed386c281 100644 --- a/theorems/T000386.md +++ b/theorems/T000386.md @@ -3,15 +3,9 @@ uid: T000386 if: and: - P000022: true - - P000162: true + - P000221: true then: P000016: true -refs: -- mathse: 4728863 - name: Compactness, pseudocompactness, and realcompactness without Hausdorff --- -Take the space $H\subseteq \mathbb R^\kappa$ (by {P162}); its projection $H_\alpha\subseteq\mathbb R$ -for each factor $\alpha<\kappa$ must be bounded (by {P22}), and thus $\overline{H_\alpha}$ is {P000016} -by the [Heine-Borel theorem](https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem). This makes $H$ -a closed subset of the {P000016} space $\prod_{\alpha<\kappa}\overline{H_\alpha}$, and thus {P000016}. +By taking Kolmogorov quotient we can assume $X$ is $T_0$. If $X\subseteq \prod_\alpha X_\alpha$ is closed where $X_\alpha$ are metric spaces, and $\pi_\alpha:X\to X_\alpha$ is the projection onto $\alpha$th coordinate, then $\pi_\alpha(X)\subseteq X_\alpha$ is {P22} and {P53}, and so {P16}. It follows that $X$ is a closed subspace of the {P16} space $\prod_\alpha \pi_\alpha(X)$, and so {P16}. From b868adbb2bd595cabd4eaed60d8a516ac53bad22 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Tue, 2 Sep 2025 10:54:40 +0200 Subject: [PATCH 07/37] changes to description, theorem to be finished --- properties/P000221.md | 6 ++++-- theorems/T000777.md | 14 ++++++++++++++ 2 files changed, 18 insertions(+), 2 deletions(-) create mode 100644 theorems/T000777.md diff --git a/properties/P000221.md b/properties/P000221.md index 4889733c13..468c3e83ca 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -19,10 +19,12 @@ refs: The topology on the space $X$ is induced by a [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness) $\mathcal{U}$. +We call uniformity $\mathcal{U}$ complete if every Cauchy filter $\mathcal{F}$ on $(X, \mathcal{U})$ converges. Equivalently, every Cauchy net $(x_i)_{i\in I}$ on $(X, \mathcal{U})$ converges. Here Cauchy filter is a filter $\mathcal{F}$ such that for every $U\in\mathcal{U}$ there exists $A\in\mathcal{F}$ such that $A\times A\subseteq U$. A Cauchy net is a net $(x_i)_{i\in I}$ such that for every $U\in\mathcal{U}$ there exists $i_0$ such that $(x_j, x_k)\in U$ for $j, k\geq i_0$. + If $X$ is $T_0$ this is equivalent to the following (see {{zb:0684.54001}} exercise 8.5.13a): -2. $X$ is a closed subspace of a product of {P55} spaces -3. $X$ is a closed subspace of a product of {P53} spaces. +1. $X$ is a closed subspace of a product of {P55} spaces +2. $X$ is a closed subspace of a product of {P53} spaces. Compare with definition in 15.7 of {{doi:10.1007/978-1-4615-7819-2}} where uniform structure is defined using pseudometrics instead. diff --git a/theorems/T000777.md b/theorems/T000777.md new file mode 100644 index 0000000000..04c6f0a1a9 --- /dev/null +++ b/theorems/T000777.md @@ -0,0 +1,14 @@ +--- +uid: T000777 +if: + and: + - P000134: true + - P000030: true +then: + P000221: true +refs: + - mr: 370454 + name: General Topology (Kelley) +--- + +To be finished. From fb3f3033f07188b2b392ddd2f1743c4648095dac Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Tue, 2 Sep 2025 11:36:35 +0200 Subject: [PATCH 08/37] proof of T777 updated --- theorems/T000386.md | 2 +- theorems/T000777.md | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/theorems/T000386.md b/theorems/T000386.md index 6ed386c281..ab53aa24d5 100644 --- a/theorems/T000386.md +++ b/theorems/T000386.md @@ -8,4 +8,4 @@ then: P000016: true --- -By taking Kolmogorov quotient we can assume $X$ is $T_0$. If $X\subseteq \prod_\alpha X_\alpha$ is closed where $X_\alpha$ are metric spaces, and $\pi_\alpha:X\to X_\alpha$ is the projection onto $\alpha$th coordinate, then $\pi_\alpha(X)\subseteq X_\alpha$ is {P22} and {P53}, and so {P16}. It follows that $X$ is a closed subspace of the {P16} space $\prod_\alpha \pi_\alpha(X)$, and so {P16}. +By taking Kolmogorov quotient we can assume $X$ is $T_0$. If $X\subseteq \prod_\alpha X_\alpha$ is closed where $X_\alpha$ are metric spaces, and $\pi_\alpha:X\to X_\alpha$ are projections, then $\pi_\alpha(X)\subseteq X_\alpha$ is {P22} and {P53}, and so {P16}. It follows that $X$ is a closed subspace of the {P16} space $\prod_\alpha \pi_\alpha(X)$, and so {P16}. diff --git a/theorems/T000777.md b/theorems/T000777.md index 04c6f0a1a9..5f3f9f8d48 100644 --- a/theorems/T000777.md +++ b/theorems/T000777.md @@ -11,4 +11,4 @@ refs: name: General Topology (Kelley) --- -To be finished. +By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207}, the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points, for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From theorem 5.28 of {{mr:370454}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. From 1fe2b16d555eae4113cc8a15d34862d81d4e0700 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Sun, 12 Jul 2026 22:27:25 +0200 Subject: [PATCH 09/37] Update properties/P000221.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000221.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000221.md b/properties/P000221.md index 468c3e83ca..f8637741ba 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -19,7 +19,7 @@ refs: The topology on the space $X$ is induced by a [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness) $\mathcal{U}$. -We call uniformity $\mathcal{U}$ complete if every Cauchy filter $\mathcal{F}$ on $(X, \mathcal{U})$ converges. Equivalently, every Cauchy net $(x_i)_{i\in I}$ on $(X, \mathcal{U})$ converges. Here Cauchy filter is a filter $\mathcal{F}$ such that for every $U\in\mathcal{U}$ there exists $A\in\mathcal{F}$ such that $A\times A\subseteq U$. A Cauchy net is a net $(x_i)_{i\in I}$ such that for every $U\in\mathcal{U}$ there exists $i_0$ such that $(x_j, x_k)\in U$ for $j, k\geq i_0$. +We call a uniformity $\mathcal{U}$ *complete* if every Cauchy filter $\mathcal{F}$ on $(X, \mathcal{U})$ converges. Equivalently, every Cauchy net $(x_i)_{i\in I}$ on $(X, \mathcal{U})$ converges. Here a *Cauchy filter* is a filter $\mathcal{F}$ such that for every $U\in\mathcal{U}$ there exists $A\in\mathcal{F}$ such that $A\times A\subseteq U$. A *Cauchy net* is a net $(x_i)_{i\in I}$ such that for every $U\in\mathcal{U}$ there exists $i_0$ such that $(x_j, x_k)\in U$ for $j, k\geq i_0$. If $X$ is $T_0$ this is equivalent to the following (see {{zb:0684.54001}} exercise 8.5.13a): From 95c5c37f8df76eb4e1b4513efebe162dfd450c11 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Sun, 12 Jul 2026 22:31:18 +0200 Subject: [PATCH 10/37] Apply suggestion from @prabau Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000221.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000221.md b/properties/P000221.md index f8637741ba..692a665e07 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -28,7 +28,7 @@ If $X$ is $T_0$ this is equivalent to the following (see {{zb:0684.54001}} exerc Compare with definition in 15.7 of {{doi:10.1007/978-1-4615-7819-2}} where uniform structure is defined using pseudometrics instead. -This property is *topologically complete* in {{mr:370454}}. +This property is usually called *Dieudonné complete* (Problem 8.5.13 in {{zb:0684.54001}}) or *completely uniformizable* (Problem 39B in {{zb:1052.54001}}). ---- #### Meta-properties From e9168d0f6dfec77b6ec0bfc85771131a785d6e21 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Sun, 12 Jul 2026 22:34:33 +0200 Subject: [PATCH 11/37] Update P000221.md --- properties/P000221.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/properties/P000221.md b/properties/P000221.md index 692a665e07..1116000fab 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -15,6 +15,8 @@ refs: name: General Topology (Kelley) - zb: "0684.54001" name: General Topology (Engelking, 1989) + - zb: "1052.54001" + name: General Topology (Willard, 1970) --- The topology on the space $X$ is induced by a [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness) $\mathcal{U}$. From 0f93f37bf32b898739fefea5855534827d415ac7 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Sun, 12 Jul 2026 22:54:03 +0200 Subject: [PATCH 12/37] Update P000221.md --- properties/P000221.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000221.md b/properties/P000221.md index 1116000fab..10b2ca438e 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -30,7 +30,7 @@ If $X$ is $T_0$ this is equivalent to the following (see {{zb:0684.54001}} exerc Compare with definition in 15.7 of {{doi:10.1007/978-1-4615-7819-2}} where uniform structure is defined using pseudometrics instead. -This property is usually called *Dieudonné complete* (Problem 8.5.13 in {{zb:0684.54001}}) or *completely uniformizable* (Problem 39B in {{zb:1052.54001}}). +This property is usually called *Dieudonné complete* (Problem 8.5.13 in {{zb:0684.54001}}) or *completely uniformizable* (Problem 39B in {{zb:1052.54001}}). It is called *topologically complete* on page 208 of {{mr:370454}} ---- #### Meta-properties From 266fc1ea5e508c513dfbeeb0cf420fb82b4eead3 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Mon, 13 Jul 2026 00:28:22 +0200 Subject: [PATCH 13/37] Update T000386.md --- theorems/T000386.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000386.md b/theorems/T000386.md index ab53aa24d5..6b8ed9fb64 100644 --- a/theorems/T000386.md +++ b/theorems/T000386.md @@ -8,4 +8,4 @@ then: P000016: true --- -By taking Kolmogorov quotient we can assume $X$ is $T_0$. If $X\subseteq \prod_\alpha X_\alpha$ is closed where $X_\alpha$ are metric spaces, and $\pi_\alpha:X\to X_\alpha$ are projections, then $\pi_\alpha(X)\subseteq X_\alpha$ is {P22} and {P53}, and so {P16}. It follows that $X$ is a closed subspace of the {P16} space $\prod_\alpha \pi_\alpha(X)$, and so {P16}. +By taking Kolmogorov quotient we can assume $X$ is $T_0$. If $X\subseteq \prod_\alpha X_\alpha$ is closed where $X_\alpha$ are metric spaces, and $\pi_\alpha:X\to X_\alpha$ are projections, then $\pi_\alpha(X)\subseteq X_\alpha$ is {P22} and {P53}, and so {P16} ([explore](https://topology.pi-base.org/spaces?q=pseudocompact+%2B+metrizable+%2B+%7Ecompact)). It follows that $X$ is a closed subspace of the {P16} space $\prod_\alpha \pi_\alpha(X)$, and so {P16}. From 9fd4fe01367f4ba239a5ac3c0304d89901613716 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Mon, 13 Jul 2026 00:44:02 +0200 Subject: [PATCH 14/37] add Kolmogorov meta to P22 --- properties/P000022.md | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/properties/P000022.md b/properties/P000022.md index 09933e63c6..f046259a06 100644 --- a/properties/P000022.md +++ b/properties/P000022.md @@ -9,3 +9,8 @@ refs: Every continuous function from the space to {S25} is bounded. Defined on page 20 of {{doi:10.1007/978-1-4612-6290-9}}. + +---- +#### Meta-properties + +- $X$ satisfies this property iff its Kolmogorov quotient $\mathrm{Kol}(X)$ does. From fe75d3841292eeb55fcecc97e374a27961232d77 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Mon, 13 Jul 2026 00:48:36 +0200 Subject: [PATCH 15/37] add explore link to T777 --- theorems/T000777.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000777.md b/theorems/T000777.md index 5f3f9f8d48..684bc738c3 100644 --- a/theorems/T000777.md +++ b/theorems/T000777.md @@ -11,4 +11,4 @@ refs: name: General Topology (Kelley) --- -By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207}, the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points, for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From theorem 5.28 of {{mr:370454}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. +By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points, for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From theorem 5.28 of {{mr:370454}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. From cec9bf612c8e9825511628cf844f322284dcc607 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Mon, 13 Jul 2026 01:07:53 +0200 Subject: [PATCH 16/37] add note to P207 --- properties/P000207.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/properties/P000207.md b/properties/P000207.md index 55e423f551..8cde8ea636 100644 --- a/properties/P000207.md +++ b/properties/P000207.md @@ -17,6 +17,8 @@ For each neighborhood $U$ of the diagonal $\Delta=\{(x,x)\mid x\in X\}$ in $X\times X$, there is a neighborhood $V$ of the diagonal such that $V\circ V\subseteq U$. +This is equivalent to the family $\mathcal{U}$ of neighborhoods of $\Delta_X$ forming a uniformity, but the topology of $X$ and that of $(X, \mathcal{U})$ do not need to agree. If $X$ is {P12} then both topologies coincide. + In Theorem 2.6 of {{doi:10.2307/1993026}} this property was shown to be equivalent to *almost $2$-fully normal*: each open cover $\mathcal U$ has an open almost $2$-star refinement $\mathcal V$, that is $\mathcal{V}$ is a refinement of $\mathcal{U}$ and for any $x, y, z$ with $y, z\in \text{St}(x, \mathcal{V})$ there exists $U\in\mathcal{U}$ with $y, z\in U$. From 23ebe1c8b338dbf979fdec804fc02bfd5a7d1034 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Mon, 13 Jul 2026 01:11:12 +0200 Subject: [PATCH 17/37] added a remark that uniformity induces topology --- theorems/T000777.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000777.md b/theorems/T000777.md index 684bc738c3..a00dc00664 100644 --- a/theorems/T000777.md +++ b/theorems/T000777.md @@ -11,4 +11,4 @@ refs: name: General Topology (Kelley) --- -By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points, for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From theorem 5.28 of {{mr:370454}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. +By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular)), the uniformity is admissible for $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points, for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From theorem 5.28 of {{mr:370454}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. From f9d4d2c83a8ea96857e1e4151a7f0e493ded5fda Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Mon, 13 Jul 2026 01:24:32 +0200 Subject: [PATCH 18/37] add explore to T776 --- theorems/T000776.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000776.md b/theorems/T000776.md index 2e5b06dd6b..b61c214efc 100644 --- a/theorems/T000776.md +++ b/theorems/T000776.md @@ -12,4 +12,4 @@ refs: name: Rings of Continuous Functions (Gillman & Jerison) --- -A {P221} {P1} space is {P6}. Now apply theorem 15.20 of {{doi:10.1007/978-1-4615-7819-2}}. +A {P221} {P1} space is {P6} ([exlore](https://topology.pi-base.org/spaces?q=dieudonne+complete+%2B+T_0+%2B+not+completely+regular)). Now apply theorem 15.20 of {{doi:10.1007/978-1-4615-7819-2}}. From ecb2f1ada576393b9011bf99e0d1b66da352f307 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Mon, 13 Jul 2026 01:48:57 +0200 Subject: [PATCH 19/37] Update theorems/T000777.md Co-authored-by: Felix Pernegger --- theorems/T000777.md | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/theorems/T000777.md b/theorems/T000777.md index a00dc00664..adb430d87c 100644 --- a/theorems/T000777.md +++ b/theorems/T000777.md @@ -11,4 +11,10 @@ refs: name: General Topology (Kelley) --- -By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular)), the uniformity is admissible for $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points, for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From theorem 5.28 of {{mr:370454}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. +By taking Kolmogorov quotient we can assume $X$ is {P3}. + +Assume $X$ is not {P221}. Since $X$ is {P207} ([Explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular)), the uniformity is admissible for $X$. + +Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points, for each $x\in X$, there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. + +From Theorem 5.28 of {{zb:1358.54001}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. From 5fd2db6ce65833f3ab3394df62eb456f7b94a16f Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Mon, 13 Jul 2026 02:20:58 +0200 Subject: [PATCH 20/37] Cauchy net converges to its adherence points --- theorems/T000777.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000777.md b/theorems/T000777.md index a00dc00664..8930a7bca2 100644 --- a/theorems/T000777.md +++ b/theorems/T000777.md @@ -11,4 +11,4 @@ refs: name: General Topology (Kelley) --- -By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular)), the uniformity is admissible for $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points, for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From theorem 5.28 of {{mr:370454}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. +By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular)), the uniformity is admissible for $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points (see theorem 6.21 of {{mr:370454}}), for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From theorem 5.28 of {{mr:370454}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. From 50036a82802db919ec195c8ce44a1c230f0cfcd1 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Mon, 13 Jul 2026 02:23:51 +0200 Subject: [PATCH 21/37] fix a broken file --- theorems/T000777.md | 4 ---- 1 file changed, 4 deletions(-) diff --git a/theorems/T000777.md b/theorems/T000777.md index 50c7ea5931..fc7d209408 100644 --- a/theorems/T000777.md +++ b/theorems/T000777.md @@ -11,14 +11,10 @@ refs: name: General Topology (Kelley) --- -<<<<<<< HEAD By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular)), the uniformity is admissible for $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points (see theorem 6.21 of {{mr:370454}}), for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From theorem 5.28 of {{mr:370454}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. -======= -By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207} ([Explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular)), the uniformity is admissible for $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points, for each $x\in X$, there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From Theorem 5.28 of {{zb:1358.54001}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. ->>>>>>> ecb2f1ada576393b9011bf99e0d1b66da352f307 From 3c2a191480a0242d676c1a111d3a01b4d642b643 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Mon, 13 Jul 2026 17:00:47 -0400 Subject: [PATCH 22/37] T777: remove duplicate text, plus cosmetic changes --- theorems/T000777.md | 11 +++++------ 1 file changed, 5 insertions(+), 6 deletions(-) diff --git a/theorems/T000777.md b/theorems/T000777.md index fc7d209408..8e39e25581 100644 --- a/theorems/T000777.md +++ b/theorems/T000777.md @@ -7,14 +7,13 @@ if: then: P000221: true refs: - - mr: 370454 + - zb: 1358.54001 name: General Topology (Kelley) --- -By taking Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular)), the uniformity is admissible for $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points (see theorem 6.21 of {{mr:370454}}), for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From theorem 5.28 of {{mr:370454}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. +By taking the Kolmogorov quotient we can assume $X$ is {P3}. +Assume $X$ is not {P221}. Since $X$ is {P207} [(Explore)](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} [(Explore)](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular), the uniformity induces the original topology of $X$. -Assume $X$ is not {P221}. Since $X$ is {P207} ([Explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal)), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} ([explore](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular)), the uniformity is admissible for $X$. +Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points (see Theorem 6.21 on page 191 of {{zb:1358.54001}}), for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. -Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points, for each $x\in X$, there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. - -From Theorem 5.28 of {{zb:1358.54001}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. +From Theorem 5.28 on page 156 of {{zb:1358.54001}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. From 11dd212032535fd6beb4fb429016d9eb49bfb446 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Mon, 13 Jul 2026 17:09:16 -0400 Subject: [PATCH 23/37] fix missing double quotes --- theorems/T000777.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000777.md b/theorems/T000777.md index 8e39e25581..cfa90618f1 100644 --- a/theorems/T000777.md +++ b/theorems/T000777.md @@ -7,7 +7,7 @@ if: then: P000221: true refs: - - zb: 1358.54001 + - zb: "1358.54001" name: General Topology (Kelley) --- From 60f499533c5fe58f3a31a2d7e1fcaf38ac7449d2 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Tue, 14 Jul 2026 17:05:13 +0200 Subject: [PATCH 24/37] Update theorems/T000777.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000777.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000777.md b/theorems/T000777.md index cfa90618f1..cedca4a6e3 100644 --- a/theorems/T000777.md +++ b/theorems/T000777.md @@ -12,7 +12,7 @@ refs: --- By taking the Kolmogorov quotient we can assume $X$ is {P3}. -Assume $X$ is not {P221}. Since $X$ is {P207} [(Explore)](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} [(Explore)](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular), the uniformity induces the original topology of $X$. +Assume $X$ is not {P221}. Since $X$ is {P207} [(Explore)](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} [(Explore)](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular), the uniformity is compatible with $X$. Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points (see Theorem 6.21 on page 191 of {{zb:1358.54001}}), for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From 710c3262f2f65b06560acd59cdbee534094558c8 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Tue, 14 Jul 2026 17:26:20 +0200 Subject: [PATCH 25/37] elaboration on nomenclature --- properties/P000055.md | 4 +--- properties/P000063.md | 6 ++++++ properties/P000221.md | 2 +- 3 files changed, 8 insertions(+), 4 deletions(-) diff --git a/properties/P000055.md b/properties/P000055.md index 5ba380a4fc..94a552027e 100644 --- a/properties/P000055.md +++ b/properties/P000055.md @@ -16,9 +16,7 @@ that is, a metric for which every Cauchy sequence converges. A sequence $(x_n)_n$ is *Cauchy* provided for each distance $\epsilon>0$, there is a natural number $N$ for which $d(x_i,x_j)<\epsilon$ for all $i,j>N$. -Defined in 24.2 of {{zb:1052.54001}}. -Defined on page 37 of {{doi:10.1007/978-1-4612-6290-9}} as -"topologically complete". +Defined in 24.2 of {{zb:1052.54001}}. Defined on page 37 of {{doi:10.1007/978-1-4612-6290-9}} as *topologically complete*. This term is also used for {P63} and {P221}. ---- #### Meta-properties diff --git a/properties/P000063.md b/properties/P000063.md index 1cbdeaf257..cf9d5efe01 100644 --- a/properties/P000063.md +++ b/properties/P000063.md @@ -1,9 +1,13 @@ --- uid: P000063 name: Čech complete +aliases: + - Topologically complete refs: - zb: "0684.54001" name: General Topology (Engelking, 1989) + - zb: "0117.15903" + name: A characterization of topologically complete spaces in the sense of E. Čech in terms of convergence of functions. (Frolik, 1963) --- A {P6} space $X$ that is a $G_\delta$ set in some compactification of $X$ (equivalently, in every compactification of $X$). @@ -11,6 +15,8 @@ Equivalently, there is a sequence $\mathcal{U}_1, \mathcal{U}_2, \dots$ of open See Section 3.9 of {{zb:0684.54001}}, specifically Theorems 3.9.1 and 3.9.2 for the equivalences above. +Defined in {{zb:0117.15903}} as *topologically complete*. This term is also used for {P55} and {P221}. + ---- #### Meta-properties diff --git a/properties/P000221.md b/properties/P000221.md index 10b2ca438e..11592e58dd 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -30,7 +30,7 @@ If $X$ is $T_0$ this is equivalent to the following (see {{zb:0684.54001}} exerc Compare with definition in 15.7 of {{doi:10.1007/978-1-4615-7819-2}} where uniform structure is defined using pseudometrics instead. -This property is usually called *Dieudonné complete* (Problem 8.5.13 in {{zb:0684.54001}}) or *completely uniformizable* (Problem 39B in {{zb:1052.54001}}). It is called *topologically complete* on page 208 of {{mr:370454}} +Such spaces are called *Dieudonné complete* (Problem 8.5.13 in {{zb:0684.54001}}), *completely uniformizable* (Problem 39B in {{zb:1052.54001}}) or *topologically complete* (Page 208 of {{mr:370454}}). The last term is also used for {P55} and {P63}. ---- #### Meta-properties From 99ed32eca14082ed82c0bf5d296d06261ddb71a0 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Tue, 14 Jul 2026 17:55:01 +0200 Subject: [PATCH 26/37] rephrased equivalent statements to include non-T0 case --- properties/P000221.md | 11 ++++++----- 1 file changed, 6 insertions(+), 5 deletions(-) diff --git a/properties/P000221.md b/properties/P000221.md index 11592e58dd..cdf908958d 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -19,16 +19,17 @@ refs: name: General Topology (Willard, 1970) --- -The topology on the space $X$ is induced by a [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness) $\mathcal{U}$. +The topology on the space $X$ is induced by a [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness). We call a uniformity $\mathcal{U}$ *complete* if every Cauchy filter $\mathcal{F}$ on $(X, \mathcal{U})$ converges. Equivalently, every Cauchy net $(x_i)_{i\in I}$ on $(X, \mathcal{U})$ converges. Here a *Cauchy filter* is a filter $\mathcal{F}$ such that for every $U\in\mathcal{U}$ there exists $A\in\mathcal{F}$ such that $A\times A\subseteq U$. A *Cauchy net* is a net $(x_i)_{i\in I}$ such that for every $U\in\mathcal{U}$ there exists $i_0$ such that $(x_j, x_k)\in U$ for $j, k\geq i_0$. -If $X$ is $T_0$ this is equivalent to the following (see {{zb:0684.54001}} exercise 8.5.13a): +The following are equivalent (compare to exercise 8.5.13a of {{zb:0684.54001}}): -1. $X$ is a closed subspace of a product of {P55} spaces -2. $X$ is a closed subspace of a product of {P53} spaces. +1. $X$ is {P221}, +2. $X$ is a closed subspace of a product of completely {P121} spaces, +3. $X$ is a closed subspace of a product of {P121} spaces. -Compare with definition in 15.7 of {{doi:10.1007/978-1-4615-7819-2}} where uniform structure is defined using pseudometrics instead. +Compare this definition of complete uniformity with the definition in 15.7 of {{doi:10.1007/978-1-4615-7819-2}} where uniform structure is defined using pseudometrics instead. Such spaces are called *Dieudonné complete* (Problem 8.5.13 in {{zb:0684.54001}}), *completely uniformizable* (Problem 39B in {{zb:1052.54001}}) or *topologically complete* (Page 208 of {{mr:370454}}). The last term is also used for {P55} and {P63}. From d3b077d65ee9a593fd02a5c3cb38f18b52f9398f Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Thu, 16 Jul 2026 22:47:24 +0200 Subject: [PATCH 27/37] renamed theorems --- theorems/{T000774.md => T000914.md} | 2 +- theorems/{T000775.md => T000915.md} | 2 +- theorems/{T000776.md => T000916.md} | 2 +- theorems/{T000777.md => T000917.md} | 2 +- 4 files changed, 4 insertions(+), 4 deletions(-) rename theorems/{T000774.md => T000914.md} (89%) rename theorems/{T000775.md => T000915.md} (96%) rename theorems/{T000776.md => T000916.md} (96%) rename theorems/{T000777.md => T000917.md} (99%) diff --git a/theorems/T000774.md b/theorems/T000914.md similarity index 89% rename from theorems/T000774.md rename to theorems/T000914.md index 84d37d06e5..46c416cc59 100644 --- a/theorems/T000774.md +++ b/theorems/T000914.md @@ -1,5 +1,5 @@ --- -uid: T000774 +uid: T000914 if: P000221: true then: diff --git a/theorems/T000775.md b/theorems/T000915.md similarity index 96% rename from theorems/T000775.md rename to theorems/T000915.md index 6551d59d05..88c6b6663c 100644 --- a/theorems/T000775.md +++ b/theorems/T000915.md @@ -1,5 +1,5 @@ --- -uid: T000775 +uid: T000915 if: P000162: true then: diff --git a/theorems/T000776.md b/theorems/T000916.md similarity index 96% rename from theorems/T000776.md rename to theorems/T000916.md index b61c214efc..37d936654f 100644 --- a/theorems/T000776.md +++ b/theorems/T000916.md @@ -1,5 +1,5 @@ --- -uid: T000776 +uid: T000916 if: and: - P000001: true diff --git a/theorems/T000777.md b/theorems/T000917.md similarity index 99% rename from theorems/T000777.md rename to theorems/T000917.md index cedca4a6e3..61e9fa1764 100644 --- a/theorems/T000777.md +++ b/theorems/T000917.md @@ -1,5 +1,5 @@ --- -uid: T000777 +uid: T000917 if: and: - P000134: true From 7d964f8ec14687c68320784cfcc497961a5d2d79 Mon Sep 17 00:00:00 2001 From: Batixx Date: Thu, 16 Jul 2026 23:03:56 +0200 Subject: [PATCH 28/37] doi => zb --- properties/P000049.md | 4 ++-- properties/P000085.md | 4 ++-- properties/P000162.md | 4 ++-- properties/P000164.md | 4 ++-- properties/P000215.md | 4 ++-- properties/P000221.md | 4 ++-- spaces/S000074/properties/P000162.md | 4 ++-- spaces/S000107/properties/P000162.md | 4 ++-- spaces/S000153/properties/P000162.md | 4 ++-- spaces/S000208/README.md | 4 ++-- spaces/S000216/README.md | 4 ++-- theorems/T000382.md | 2 +- theorems/T000383.md | 4 ++-- theorems/T000384.md | 4 ++-- theorems/T000742.md | 4 ++-- theorems/T000915.md | 4 ++-- theorems/T000916.md | 4 ++-- 17 files changed, 33 insertions(+), 33 deletions(-) diff --git a/properties/P000049.md b/properties/P000049.md index a54e17224f..2729d11097 100644 --- a/properties/P000049.md +++ b/properties/P000049.md @@ -4,7 +4,7 @@ name: Extremally disconnected refs: - zb: "1052.54001" name: General Topology (Willard) - - doi: 10.1007/978-1-4615-7819-2 + - zb: "1380.46022" name: Rings of Continuous Functions (Gillman & Jerison) - zb: "0386.54001" name: Counterexamples in Topology @@ -18,7 +18,7 @@ The closure of every open set in $X$ is open or, equivalently, clopen. Equivalently, any two disjoint open sets have disjoint closures. -Defined in problem 15G of {{zb:1052.54001}} and problem 1H of {{doi:10.1007/978-1-4615-7819-2}}. +Defined in problem 15G of {{zb:1052.54001}} and problem 1H of {{zb:1380.46022}}. {{zb:0386.54001}} defines it on page 32 with the additional assumption of {P3}, which we do not assume here. diff --git a/properties/P000085.md b/properties/P000085.md index d89d7e236a..13ad2b4191 100644 --- a/properties/P000085.md +++ b/properties/P000085.md @@ -2,7 +2,7 @@ uid: P000085 name: Basically disconnected refs: - - doi: 10.1007/978-1-4615-7819-2 + - zb: "1380.46022" name: Rings of Continuous Functions (Gillman & Jerison) --- @@ -13,7 +13,7 @@ equivalently, the complement of a zero set. Equivalently, any two disjoint open sets, at least one of which is a cozero set, have disjoint closures. -Defined in problem 1H of {{doi:10.1007/978-1-4615-7819-2}}. +Defined in problem 1H of {{zb:1380.46022}}. No additional separation axiom is assumed here. diff --git a/properties/P000162.md b/properties/P000162.md index 813b9a91f8..5ec4567f0c 100644 --- a/properties/P000162.md +++ b/properties/P000162.md @@ -4,7 +4,7 @@ name: Realcompact refs: - zb: "0684.54001" name: General Topology (Engelking, 1989) - - doi: 10.1007/978-1-4615-7819-2 + - zb: "1380.46022" name: Rings of Continuous Functions (Gillman and Jerison) - wikipedia: Ultrafilter name: Ultrafilter on Wikipedia @@ -12,7 +12,7 @@ refs: A space $X$ that is homeomorphic to a closed subset of $\mathbb{R}^\kappa$ for some cardinal $\kappa$. -Equivalently (see {{doi:10.1007/978-1-4615-7819-2}}), $X$ is {P6} and every real $z$-ultrafilter $\mathcal U$ on the space $X$ is fixed, +Equivalently (see {{zb:1380.46022}}), $X$ is {P6} and every real $z$-ultrafilter $\mathcal U$ on the space $X$ is fixed, that is, $\bigcap\mathcal{U}\neq\emptyset$. A *$z$-ultrafilter* is an ultrafilter on the lattice of zero-sets of $X$ (see {{wikipedia:Ultrafilter}} for the general definition of an ultrafilter on a poset). A *real $z$-ultrafilter* is a $z$-ultrafilter with countable intersection property, that is, for any countable $\mathcal{F}\subseteq \mathcal{U}$ we have $\bigcap\mathcal{F}\neq \emptyset$. diff --git a/properties/P000164.md b/properties/P000164.md index af5a145864..660f996937 100644 --- a/properties/P000164.md +++ b/properties/P000164.md @@ -6,7 +6,7 @@ aliases: refs: - wikipedia: Measurable_cardinal name: Measurable cardinal on Wikipedia -- doi: 10.1007/978-1-4615-7819-2 +- zb: "1380.46022" name: Rings of Continuous Functions (Gillman and Jerison) - doi: 10.1007/3-540-44761-X name: Set Theory (Jech) @@ -22,7 +22,7 @@ A cardinal $\kappa$ is called *measurable* if $\kappa$ is uncountable and there Equivalently, $\kappa$ is uncountable and there exists a free ultrafilter $\mathcal{U}$ on $\kappa$ such that $\mathcal{U}$ is *$\kappa$-complete*, i.e., if $\mathcal{F}\subseteq \mathcal{U}$ and $|\mathcal{F}| < \kappa$ then $\bigcap\mathcal{F}\in \mathcal{U}$. (See {{wikipedia:Measurable_cardinal}} for more details.) -Note: Some authors, for example {{doi:10.1007/978-1-4615-7819-2}}, refer to measurable cardinals as those cardinals $\kappa$ for which there exists a $\sigma$-additive measure $\mu:2^\kappa\to \{0, 1\}$ which is non-trivial. If $\kappa$ is the smallest such cardinal, then a non-trivial $\sigma$-additive measure $\mu:2^\kappa\to \{0, 1\}$ is $\kappa$-additive (see lemma 10.2 of {{doi:10.1007/3-540-44761-X}} and comments preceding it), so $\kappa$ is also measurable by the above definition. +Note: Some authors, for example {{zb:1380.46022}}, refer to measurable cardinals as those cardinals $\kappa$ for which there exists a $\sigma$-additive measure $\mu:2^\kappa\to \{0, 1\}$ which is non-trivial. If $\kappa$ is the smallest such cardinal, then a non-trivial $\sigma$-additive measure $\mu:2^\kappa\to \{0, 1\}$ is $\kappa$-additive (see lemma 10.2 of {{doi:10.1007/3-540-44761-X}} and comments preceding it), so $\kappa$ is also measurable by the above definition. (The existence of a measurable cardinal cannot be proven in ZFC. So spaces whose construction does not depend on set-theoretic axioms beyond ZFC should never have this property marked as false.) diff --git a/properties/P000215.md b/properties/P000215.md index 8b8ed9bebc..47d46163f0 100644 --- a/properties/P000215.md +++ b/properties/P000215.md @@ -2,13 +2,13 @@ uid: P000215 name: Hereditarily realcompact refs: -- doi: 10.1007/978-1-4615-7819-2 +- zb: "1380.46022" name: Rings of Continuous Functions (Gillman & Jerison) --- Every subspace is {P162}. -Equivalently, $X$ is {P6} and $X\setminus \{x\}$ is realcompact for each $x\in X$. (theorem 8.17 of {{doi:10.1007/978-1-4615-7819-2}}) +Equivalently, $X$ is {P6} and $X\setminus \{x\}$ is realcompact for each $x\in X$. (theorem 8.17 of {{zb:1380.46022}}) ---- #### Meta-properties diff --git a/properties/P000221.md b/properties/P000221.md index cdf908958d..c2741bf7a8 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -9,7 +9,7 @@ refs: name: Uniform space on Wikipedia - wikipedia: Completely_uniformizable_space name: Completely uniformizable space - - doi: 10.1007/978-1-4615-7819-2 + - zb: "1380.46022" name: Rings of Continuous Functions (Gillman & Jerison) - mr: 370454 name: General Topology (Kelley) @@ -29,7 +29,7 @@ The following are equivalent (compare to exercise 8.5.13a of {{zb:0684.54001}}): 2. $X$ is a closed subspace of a product of completely {P121} spaces, 3. $X$ is a closed subspace of a product of {P121} spaces. -Compare this definition of complete uniformity with the definition in 15.7 of {{doi:10.1007/978-1-4615-7819-2}} where uniform structure is defined using pseudometrics instead. +Compare this definition of complete uniformity with the definition in 15.7 of {{zb:1380.46022}} where uniform structure is defined using pseudometrics instead. Such spaces are called *Dieudonné complete* (Problem 8.5.13 in {{zb:0684.54001}}), *completely uniformizable* (Problem 39B in {{zb:1052.54001}}) or *topologically complete* (Page 208 of {{mr:370454}}). The last term is also used for {P55} and {P63}. diff --git a/spaces/S000074/properties/P000162.md b/spaces/S000074/properties/P000162.md index e8fb1a46bd..563d3d56c1 100644 --- a/spaces/S000074/properties/P000162.md +++ b/spaces/S000074/properties/P000162.md @@ -3,8 +3,8 @@ space: S000074 property: P000162 value: true refs: -- doi: 10.1007/978-1-4615-7819-2 +- zb: "1380.46022" name: Rings of Continuous Functions (Gillman and Jerison) --- -The identity function $\text{Id}:X\to \mathbb{R}^2$, $\text{Id}(x) = x$ from {S74} to {S176} is a continuous injection. Since every subspace of {S176} is realcompact ({S176} is {P5} and {P131}, and see {T384}), {S74} is realcompact from corollary 8.18 in {{doi:10.1007/978-1-4615-7819-2}}. +The identity function $\text{Id}:X\to \mathbb{R}^2$, $\text{Id}(x) = x$ from {S74} to {S176} is a continuous injection. Since every subspace of {S176} is realcompact ({S176} is {P5} and {P131}, and see {T384}), {S74} is realcompact from corollary 8.18 in {{zb:1380.46022}}. diff --git a/spaces/S000107/properties/P000162.md b/spaces/S000107/properties/P000162.md index 223c11d460..c6ef1b0d2d 100644 --- a/spaces/S000107/properties/P000162.md +++ b/spaces/S000107/properties/P000162.md @@ -3,9 +3,9 @@ space: S000107 property: P000162 value: true refs: -- doi: 10.1007/978-1-4615-7819-2 +- zb: "1380.46022" name: Rings of Continuous Functions (Gillman and Jerison) --- -The identity function $\text{Id}:\mathbb{R}^\omega\to \mathbb{R}^\omega$, $\text{Id}(x) = x$ from {S107} to $\mathbb{R}^\omega$ with product topology is a continuous bijection. Since every subspace of $\mathbb{R}^\omega$ with product topology is realcompact ($\mathbb{R}^\omega$ is {P5} and {P131}, and see {T384}), {S107} is realcompact from corollary 8.18 in {{doi:10.1007/978-1-4615-7819-2}}. +The identity function $\text{Id}:\mathbb{R}^\omega\to \mathbb{R}^\omega$, $\text{Id}(x) = x$ from {S107} to $\mathbb{R}^\omega$ with product topology is a continuous bijection. Since every subspace of $\mathbb{R}^\omega$ with product topology is realcompact ($\mathbb{R}^\omega$ is {P5} and {P131}, and see {T384}), {S107} is realcompact from corollary 8.18 in {{zb:1380.46022}}. diff --git a/spaces/S000153/properties/P000162.md b/spaces/S000153/properties/P000162.md index 58cc0fca05..d35ecb5671 100644 --- a/spaces/S000153/properties/P000162.md +++ b/spaces/S000153/properties/P000162.md @@ -3,8 +3,8 @@ space: S000153 property: P000162 value: false refs: -- doi: 10.1007/978-1-4615-7819-2 +- zb: "1380.46022" name: Rings of Continuous Functions (Gillman and Jerison) --- -The subspace $X\subseteq Y$ of {S153} given by $X = \{(x, 0) : 0 < x < \omega_1\}$ is a closed copy of $\omega_1$ in $Y$. If $Y$ were realcompact, then its closed subspace $X$ would be realcompact (see theorem 8.10 in {{doi:10.1007/978-1-4615-7819-2}}). But $\omega_1$ is a pseudocompact non-compact Tychonoff space, so is not realcompact, see {T388}. +The subspace $X\subseteq Y$ of {S153} given by $X = \{(x, 0) : 0 < x < \omega_1\}$ is a closed copy of $\omega_1$ in $Y$. If $Y$ were realcompact, then its closed subspace $X$ would be realcompact (see theorem 8.10 in {{zb:1380.46022}}). But $\omega_1$ is a pseudocompact non-compact Tychonoff space, so is not realcompact, see {T388}. diff --git a/spaces/S000208/README.md b/spaces/S000208/README.md index 912e13430c..8032cbb16c 100644 --- a/spaces/S000208/README.md +++ b/spaces/S000208/README.md @@ -4,7 +4,7 @@ name: Hewitt realcompactification of Rudin's Dowker space refs: - zb: "0224.54019" name: A normal space X for which X×I is not normal (M.E. Rudin) - - doi: 10.1007/978-1-4615-7819-2 + - zb: "1380.46022" name: Rings of Continuous Functions (Gillman & Jerison) - wikipedia: Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets name: Cofinality on Wikipedia @@ -13,4 +13,4 @@ refs: $X$ is the subspace of the product $\prod_{n\in\omega}(\omega_{n+1}+1)$ with the box topology consisting of all $f\in \prod_{n\in \omega}(\omega_{n+1}+1)$ such that $\omega< \text{cf}(f(n))$ for all $n$ (see {{wikipedia:Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets}}). Defined (as the space called $X'$) and shown to be the Hewitt realcompactification of {S138} in section IV.4 of {{zb:0224.54019}} -(see remark 8.8 of {{doi:10.1007/978-1-4615-7819-2}} for the definition of Hewitt realcompactification). +(see remark 8.8 of {{zb:1380.46022}} for the definition of Hewitt realcompactification). diff --git a/spaces/S000216/README.md b/spaces/S000216/README.md index 87ada22090..fe6a2a565b 100644 --- a/spaces/S000216/README.md +++ b/spaces/S000216/README.md @@ -2,7 +2,7 @@ uid: S000216 name: Katětov's non-normal subspace of $\beta\mathbb{N}$ refs: - - doi: 10.1007/978-1-4615-7819-2 + - zb: "1380.46022" name: Rings of Continuous Functions (Gillman & Jerison) --- @@ -10,4 +10,4 @@ Fix a bijection $\varphi:\mathbb{N}\to\mathbb{Q}$. For each irrational $r$ fix a Katětov's non-normal subspace of $\beta\mathbb{N}$ is the space $X=\mathbb{N}\cup D$ where $D = \{p_E : E\in\mathcal{E}\}$. -Constructed in exercise 6Q of {{doi:10.1007/978-1-4615-7819-2}}. +Constructed in exercise 6Q of {{zb:1380.46022}}. diff --git a/theorems/T000382.md b/theorems/T000382.md index bef4bd18ff..a183dffa13 100644 --- a/theorems/T000382.md +++ b/theorems/T000382.md @@ -10,7 +10,7 @@ then: refs: - doi: 10.1090/S0002-9939-1973-0322812-9 name: Certain Subsets of Products of θ-refinable Spaces are Realcompact (P. Zenor) - - doi: 10.1007/978-1-4615-7819-2 + - zb: "1380.46022" name: Rings of Continuous Functions (Gillman & Jerison) --- diff --git a/theorems/T000383.md b/theorems/T000383.md index 8c3adff918..b2a939122b 100644 --- a/theorems/T000383.md +++ b/theorems/T000383.md @@ -5,12 +5,12 @@ if: then: P000164: true refs: - - doi: 10.1007/978-1-4615-7819-2 + - zb: "1380.46022" name: Rings of Continuous Functions (Gillman and Jerison) - wikipedia: Measurable_cardinal name: Measurable cardinal on Wikipedia --- -See Theorem 12.5 in {{doi:10.1007/978-1-4615-7819-2}}: in ZFC a measurable cardinal must be strongly inaccessible. +See Theorem 12.5 in {{zb:1380.46022}}: in ZFC a measurable cardinal must be strongly inaccessible. Also {{wikipedia:Measurable_cardinal}}. diff --git a/theorems/T000384.md b/theorems/T000384.md index 67e0880e0a..82a0a59278 100644 --- a/theorems/T000384.md +++ b/theorems/T000384.md @@ -9,10 +9,10 @@ then: refs: - zb: "0684.54001" name: General Topology (Engelking, 1989) -- doi: 10.1007/978-1-4615-7819-2 +- zb: "1380.46022" name: Rings of Continuous Functions (Gillman and Jerison) --- See Theorem 3.8.2 of {{zb:0684.54001}} (where the {P18} property assumes {P5}). -Also Theorem 8.2 of {{doi:10.1007/978-1-4615-7819-2}} (where all spaces are assumed {P6}). +Also Theorem 8.2 of {{zb:1380.46022}} (where all spaces are assumed {P6}). diff --git a/theorems/T000742.md b/theorems/T000742.md index 5f125f830d..71020d8140 100644 --- a/theorems/T000742.md +++ b/theorems/T000742.md @@ -8,7 +8,7 @@ if: then: P000215: true refs: -- doi: 10.1007/978-1-4615-7819-2 +- zb: "1380.46022" name: Rings of Continuous Functions (Gillman & Jerison) --- @@ -17,5 +17,5 @@ Every {P53} space is {P7} and {P194} [(Explore)](https://topology.pi-base.org/spaces?q=Metrizable%2B%7ESubmetacompact). So every subspace of $Y$ satisfies the hypotheses of {T382}, and hence is {P162}. -By Corollary 8.18 of {{doi:10.1007/978-1-4615-7819-2}} +By Corollary 8.18 of {{zb:1380.46022}} every subspace of $X$ is {P162}. diff --git a/theorems/T000915.md b/theorems/T000915.md index 88c6b6663c..39abd84e90 100644 --- a/theorems/T000915.md +++ b/theorems/T000915.md @@ -5,9 +5,9 @@ if: then: P000221: true refs: - - doi: 10.1007/978-1-4615-7819-2 + - zb: "1380.46022" name: Rings of Continuous Functions (Gillman & Jerison) --- -See corollary 15.14 of {{doi:10.1007/978-1-4615-7819-2}} for complete uniformity on a {P162} space. +See corollary 15.14 of {{zb:1380.46022}} for complete uniformity on a {P162} space. Alternatively, a {P162} space is a closed subspace of product of {S25} and {S25|P53}. diff --git a/theorems/T000916.md b/theorems/T000916.md index 37d936654f..eb85b693c3 100644 --- a/theorems/T000916.md +++ b/theorems/T000916.md @@ -8,8 +8,8 @@ if: then: P000162: true refs: - - doi: 10.1007/978-1-4615-7819-2 + - zb: "1380.46022" name: Rings of Continuous Functions (Gillman & Jerison) --- -A {P221} {P1} space is {P6} ([exlore](https://topology.pi-base.org/spaces?q=dieudonne+complete+%2B+T_0+%2B+not+completely+regular)). Now apply theorem 15.20 of {{doi:10.1007/978-1-4615-7819-2}}. +A {P221} {P1} space is {P6} ([exlore](https://topology.pi-base.org/spaces?q=dieudonne+complete+%2B+T_0+%2B+not+completely+regular)). Now apply theorem 15.20 of {{zb:1380.46022}}. From f26685d5d4ebb1848411dad969bc9c96ee1911bc Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Fri, 17 Jul 2026 00:07:16 -0400 Subject: [PATCH 29/37] P63 cosmetic --- properties/P000063.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/properties/P000063.md b/properties/P000063.md index cf9d5efe01..a4b0f5e212 100644 --- a/properties/P000063.md +++ b/properties/P000063.md @@ -13,9 +13,10 @@ A {P6} space $X$ that is a $G_\delta$ set in some compactification of $X$ (equiv Equivalently, there is a sequence $\mathcal{U}_1, \mathcal{U}_2, \dots$ of open covers of $X$ such that whenever $\mathcal{F}$ is a family of closed sets with the finite intersection property and such that for each $n$ there is some $F_n \in \mathcal{F}$ with $F_n \subseteq U$ for some $U \in \mathcal{U}_n$, then $\bigcap \mathcal F \neq \emptyset$. -See Section 3.9 of {{zb:0684.54001}}, specifically Theorems 3.9.1 and 3.9.2 for the equivalences above. +See Section 3.9 ("Čech-complete spaces") of {{zb:0684.54001}}, specifically Theorems 3.9.1 and 3.9.2 for the equivalences above. -Defined in {{zb:0117.15903}} as *topologically complete*. This term is also used for {P55} and {P221}. +Such spaces are called *topologically complete* in {{zb:0117.15903}}; +this last term has also been used for {P55} and {P221}. ---- #### Meta-properties From 9517dd62a368bbbf09cb98e57c6d9cde5414471d Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Fri, 17 Jul 2026 02:24:55 -0400 Subject: [PATCH 30/37] P221 updates --- properties/P000221.md | 26 +++++++++++++++----------- 1 file changed, 15 insertions(+), 11 deletions(-) diff --git a/properties/P000221.md b/properties/P000221.md index c2741bf7a8..249c988a62 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -5,8 +5,6 @@ aliases: - Completely uniformizable - Topologically complete refs: - - wikipedia: Uniform_space - name: Uniform space on Wikipedia - wikipedia: Completely_uniformizable_space name: Completely uniformizable space - zb: "1380.46022" @@ -17,23 +15,29 @@ refs: name: General Topology (Engelking, 1989) - zb: "1052.54001" name: General Topology (Willard, 1970) + - zb: "0024.36301" + name: Sur les espaces uniformes complets (Dieudonné) --- -The topology on the space $X$ is induced by a [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness). +The topology on $X$ is induced by a [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness). -We call a uniformity $\mathcal{U}$ *complete* if every Cauchy filter $\mathcal{F}$ on $(X, \mathcal{U})$ converges. Equivalently, every Cauchy net $(x_i)_{i\in I}$ on $(X, \mathcal{U})$ converges. Here a *Cauchy filter* is a filter $\mathcal{F}$ such that for every $U\in\mathcal{U}$ there exists $A\in\mathcal{F}$ such that $A\times A\subseteq U$. A *Cauchy net* is a net $(x_i)_{i\in I}$ such that for every $U\in\mathcal{U}$ there exists $i_0$ such that $(x_j, x_k)\in U$ for $j, k\geq i_0$. - -The following are equivalent (compare to exercise 8.5.13a of {{zb:0684.54001}}): +This is equivalent to each of the following: +- $X$ is homeomorphic to a closed subspace of a product of completely pseudometrizable spaces. +- $X$ is homeomorphic to a closed subspace of a product of {P121} spaces. -1. $X$ is {P221}, -2. $X$ is a closed subspace of a product of completely {P121} spaces, -3. $X$ is a closed subspace of a product of {P121} spaces. +For the equivalence, see Problem 8.5.13(z) in {{zb:0684.54001}} and page 285 of {{zb:0024.36301}}. -Compare this definition of complete uniformity with the definition in 15.7 of {{zb:1380.46022}} where uniform structure is defined using pseudometrics instead. +Terminology: +We call a uniformity $\mathcal{U}$ *complete* if every Cauchy filter $\mathcal{F}$ on $(X, \mathcal{U})$ converges. Equivalently, every Cauchy net $(x_i)_{i\in I}$ on $(X, \mathcal{U})$ converges. Here a *Cauchy filter* is a filter $\mathcal{F}$ such that for every $U\in\mathcal{U}$ there exists $A\in\mathcal{F}$ such that $A\times A\subseteq U$. A *Cauchy net* is a net $(x_i)_{i\in I}$ such that for every $U\in\mathcal{U}$ there exists $i_0$ such that $(x_j, x_k)\in U$ for $j, k\geq i_0$. +(Compare with the definition of complete uniformity in 15.7 of {{zb:1380.46022}} where uniform structure is defined using pseudometrics.) -Such spaces are called *Dieudonné complete* (Problem 8.5.13 in {{zb:0684.54001}}), *completely uniformizable* (Problem 39B in {{zb:1052.54001}}) or *topologically complete* (Page 208 of {{mr:370454}}). The last term is also used for {P55} and {P63}. +Such spaces are called *Dieudonné complete* (Problem 8.5.13 in {{zb:0684.54001}}) or *completely uniformizable* (Problem 39B in {{zb:1052.54001}}). +They are called *topologically complete* on page 208 of {{mr:370454}}; +this last term has also been used for {P55} and {P63}. ---- #### Meta-properties - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does. +- This property is preserved by arbitrary products. +- This property is hereditary with respect to closed sets. From ed83bb5a5ca22f20fe4d2aa937fcb13cdadf4910 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Fri, 17 Jul 2026 09:17:02 +0200 Subject: [PATCH 31/37] Update properties/P000055.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000055.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/properties/P000055.md b/properties/P000055.md index b56c7d1576..8ac9097f37 100644 --- a/properties/P000055.md +++ b/properties/P000055.md @@ -16,7 +16,9 @@ that is, a metric for which every Cauchy sequence converges. A sequence $(x_n)_n$ is *Cauchy* provided for each distance $\epsilon>0$, there is a natural number $N$ for which $d(x_i,x_j)<\epsilon$ for all $i,j>N$. -Defined in 24.2 of {{zb:1052.54001}}. Defined on page 37 of {{zb:0386.54001}} as *topologically complete*. This term is also used for {P63} and {P221}. +Defined in 24.2 of {{zb:1052.54001}}. +Called *topologically complete* on page 37 of {{zb:0386.54001}}; +this last term has also been used for {P63} and {P221}. ---- #### Meta-properties From 5f9e3d12bf8e319d8681bb7158d312ec5c908f1e Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Sat, 18 Jul 2026 01:40:04 -0400 Subject: [PATCH 32/37] T915 cosmetic --- theorems/T000915.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000915.md b/theorems/T000915.md index 39abd84e90..03e4fd95a4 100644 --- a/theorems/T000915.md +++ b/theorems/T000915.md @@ -9,5 +9,5 @@ refs: name: Rings of Continuous Functions (Gillman & Jerison) --- -See corollary 15.14 of {{zb:1380.46022}} for complete uniformity on a {P162} space. +See Corollary 15.14 of {{zb:1380.46022}} for complete uniformity on a {P162} space. Alternatively, a {P162} space is a closed subspace of product of {S25} and {S25|P53}. From d94dcda80b30ca5ec6c2a64ac7450b62fa7e71ac Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Tue, 21 Jul 2026 01:08:51 +0200 Subject: [PATCH 33/37] Update theorems/T000917.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000917.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000917.md b/theorems/T000917.md index 61e9fa1764..02d25fc229 100644 --- a/theorems/T000917.md +++ b/theorems/T000917.md @@ -14,6 +14,6 @@ refs: By taking the Kolmogorov quotient we can assume $X$ is {P3}. Assume $X$ is not {P221}. Since $X$ is {P207} [(Explore)](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+strongly+collectionwise+normal), the neighbourhoods of the diagonal $\Delta_X\subseteq X\times X$ form a uniformity $\mathcal{U}$ on $X$, and since $X$ is {P12} [(Explore)](https://topology.pi-base.org/spaces?q=R_1+%2B+paracompact+%2B+not+completely+regular), the uniformity is compatible with $X$. -Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its adherence points (see Theorem 6.21 on page 191 of {{zb:1358.54001}}), for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. +Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its cluster points (see Theorem 6.21 on page 191 of {{zb:1358.54001}}), for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. From Theorem 5.28 on page 156 of {{zb:1358.54001}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. From eaffea71bfd4f4b050c0fd6255f347c6d426d52e Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Tue, 21 Jul 2026 01:09:09 +0200 Subject: [PATCH 34/37] Update theorems/T000917.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000917.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000917.md b/theorems/T000917.md index 02d25fc229..586b66edd0 100644 --- a/theorems/T000917.md +++ b/theorems/T000917.md @@ -16,4 +16,4 @@ Assume $X$ is not {P221}. Since $X$ is {P207} [(Explore)](https://topology.pi-ba Equip $X$ with this uniformity and let $(x_i)_{i\in I}$ be a Cauchy net on $X$ that isn't convergent. Since a Cauchy net converges to each of its cluster points (see Theorem 6.21 on page 191 of {{zb:1358.54001}}), for each $x\in X$ there exists a neighbourhood $U_x$ of $x$ such that $x_i\notin U_x$ for large enough $i$. -From Theorem 5.28 on page 156 of {{zb:1358.54001}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. +From Theorem 5.28 on page 156 of {{zb:1358.54001}}, the open cover $\{U_x : x\in X\}$ is even, so there exists $V\in\mathcal{U}$ such that each $V[x] = \{y\in X :(x, y)\in V\}$ is contained in $U_z$ for some $z\in X$. If $i_0$ is such that $(x_j, x_k)\in V$ for $j, k\geq i_0$, then $(x_{i_0}, x_i)\in V$ for all $i\geq i_0$, so $x_i\in V[x_{i_0}]\subseteq U_z$ for all $i\geq i_0$. This is a contradiction since $x_i\notin U_z$ for big enough $i$. From a95d2c57830509c3a46703f85e5696ced96b9390 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Tue, 21 Jul 2026 01:09:36 +0200 Subject: [PATCH 35/37] Update properties/P000221.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000221.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/properties/P000221.md b/properties/P000221.md index 249c988a62..be383c61fe 100644 --- a/properties/P000221.md +++ b/properties/P000221.md @@ -19,7 +19,8 @@ refs: name: Sur les espaces uniformes complets (Dieudonné) --- -The topology on $X$ is induced by a [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness). +There exists at least one [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness) +that induces the topology of $X$. This is equivalent to each of the following: - $X$ is homeomorphic to a closed subspace of a product of completely pseudometrizable spaces. From 09b8ea2a8caca7fb95a2d3b1bef903a1ab82d6c1 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Tue, 21 Jul 2026 01:09:50 +0200 Subject: [PATCH 36/37] Update theorems/T000916.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000916.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/theorems/T000916.md b/theorems/T000916.md index eb85b693c3..41971d176d 100644 --- a/theorems/T000916.md +++ b/theorems/T000916.md @@ -12,4 +12,5 @@ refs: name: Rings of Continuous Functions (Gillman & Jerison) --- -A {P221} {P1} space is {P6} ([exlore](https://topology.pi-base.org/spaces?q=dieudonne+complete+%2B+T_0+%2B+not+completely+regular)). Now apply theorem 15.20 of {{zb:1380.46022}}. +A {P221} {P1} space is {P6} [(Explore)](https://topology.pi-base.org/spaces?q=dieudonne+complete+%2B+T_0+%2B+not+completely+regular). +Now apply Theorem 15.20 of {{zb:1380.46022}}. From 070ab95e7d837e74d2e487acb681db26317f632a Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Tue, 21 Jul 2026 01:10:08 +0200 Subject: [PATCH 37/37] Update theorems/T000386.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000386.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000386.md b/theorems/T000386.md index 6b8ed9fb64..bc9546b5d5 100644 --- a/theorems/T000386.md +++ b/theorems/T000386.md @@ -8,4 +8,4 @@ then: P000016: true --- -By taking Kolmogorov quotient we can assume $X$ is $T_0$. If $X\subseteq \prod_\alpha X_\alpha$ is closed where $X_\alpha$ are metric spaces, and $\pi_\alpha:X\to X_\alpha$ are projections, then $\pi_\alpha(X)\subseteq X_\alpha$ is {P22} and {P53}, and so {P16} ([explore](https://topology.pi-base.org/spaces?q=pseudocompact+%2B+metrizable+%2B+%7Ecompact)). It follows that $X$ is a closed subspace of the {P16} space $\prod_\alpha \pi_\alpha(X)$, and so {P16}. +By taking Kolmogorov quotient we can assume $X$ is $T_0$. If $X\subseteq \prod_\alpha X_\alpha$ is closed where $X_\alpha$ are metric spaces, and $\pi_\alpha:X\to X_\alpha$ are projections, then $\pi_\alpha(X)\subseteq X_\alpha$ is {P22} and {P53}, and so {P16} [(Explore)](https://topology.pi-base.org/spaces?q=pseudocompact+%2B+metrizable+%2B+%7Ecompact). It follows that $X$ is a closed subspace of the {P16} space $\prod_\alpha \pi_\alpha(X)$, and so {P16}.