From a2b9243b127e306082336a783391e65baeb21bc3 Mon Sep 17 00:00:00 2001 From: Batixx Date: Thu, 12 Mar 2026 03:11:49 +0100 Subject: [PATCH 1/6] open path components --- properties/P000234.md | 29 +++++++++++++++++++++++++++++ theorems/T000108.md | 7 +------ theorems/T000862.md | 9 +++++++++ theorems/T000863.md | 9 +++++++++ theorems/T000864.md | 9 +++++++++ 5 files changed, 57 insertions(+), 6 deletions(-) create mode 100644 properties/P000234.md create mode 100644 theorems/T000862.md create mode 100644 theorems/T000863.md create mode 100644 theorems/T000864.md diff --git a/properties/P000234.md b/properties/P000234.md new file mode 100644 index 000000000..63b7e7a25 --- /dev/null +++ b/properties/P000234.md @@ -0,0 +1,29 @@ +--- +uid: P000234 +name: Has open connected components +refs: + - mathse: 3002235 + name: Answer to "Definition of locally pathwise connected" +--- + +Every connected component of $X$ is open. + +This is equivalent to each of the following: + +* Each point $x\in X$ has a connected neighborhood. + +* Each point $x\in X$ has an open connected neighborhood. + +* $X$ is a topological disjoint union of {P36} spaces. + +The equivalence between the various conditions can be shown by arguments similar to those used in the definition of {P42}, in particular {{mathse:3002235}}. + +Compare with {P41}. + +---- +#### Meta-properties + +- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does. +- This property is hereditary with respect to clopen sets. +- This property is preserved by finite products. +- This property is preserved by arbitrary disjoint unions. diff --git a/theorems/T000108.md b/theorems/T000108.md index 2249e517b..990e0d7f7 100644 --- a/theorems/T000108.md +++ b/theorems/T000108.md @@ -3,14 +3,9 @@ uid: T000108 if: and: - P000047: true - - P000041: true + - P000234: true then: P000052: true -refs: -- zb: "0386.54001" - name: Counterexamples in Topology --- If every point has a connected neighborhood and the only connected sets are single points, then every point has a neighborhood consisting of the point itself. - -Proven on page 32 of {{zb:0386.54001}}. diff --git a/theorems/T000862.md b/theorems/T000862.md new file mode 100644 index 000000000..4b0d8d965 --- /dev/null +++ b/theorems/T000862.md @@ -0,0 +1,9 @@ +--- +uid: T000862 +if: + P000233: true +then: + P000234: true +--- + +Immediate from the definitions using {T40}. diff --git a/theorems/T000863.md b/theorems/T000863.md new file mode 100644 index 000000000..fd9df3cc4 --- /dev/null +++ b/theorems/T000863.md @@ -0,0 +1,9 @@ +--- +uid: T000863 +if: + P000041: true +then: + P000234: true +--- + +Immediate from the definitions. diff --git a/theorems/T000864.md b/theorems/T000864.md new file mode 100644 index 000000000..f685a2cb7 --- /dev/null +++ b/theorems/T000864.md @@ -0,0 +1,9 @@ +--- +uid: T000860 +if: + P000036: true +then: + P000234: true +--- + +Immediate from the definitions. From e39b702cfde16ddd4d9fc31ef7cd10339096d90e Mon Sep 17 00:00:00 2001 From: Batixx Date: Thu, 12 Mar 2026 03:16:36 +0100 Subject: [PATCH 2/6] fix uid --- theorems/T000864.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000864.md b/theorems/T000864.md index f685a2cb7..a6703884a 100644 --- a/theorems/T000864.md +++ b/theorems/T000864.md @@ -1,5 +1,5 @@ --- -uid: T000860 +uid: T000864 if: P000036: true then: From 2a2b034d7f09ddd48d5d505eca0416407e84f1e5 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Thu, 12 Mar 2026 03:31:14 +0100 Subject: [PATCH 3/6] add small clarification --- theorems/T000862.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000862.md b/theorems/T000862.md index 4b0d8d965..f7aef0b55 100644 --- a/theorems/T000862.md +++ b/theorems/T000862.md @@ -6,4 +6,4 @@ then: P000234: true --- -Immediate from the definitions using {T40}. +Immediate from the (equivalent) definitions using {T40}. From 847249c11b104febf8b5d52cda1469fd8da67de8 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Thu, 12 Mar 2026 23:48:30 -0400 Subject: [PATCH 4/6] P233: accessible reference --- properties/P000233.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/properties/P000233.md b/properties/P000233.md index ad53cc590..36cf4a2ce 100644 --- a/properties/P000233.md +++ b/properties/P000233.md @@ -20,7 +20,9 @@ This is equivalent to each of the following: The equivalence between the various conditions can be shown by arguments similar to those used in the definition of {P42}, in particular {{mathse:3002235}}. -This property is used in the literature, but usually without a separate name. One exception is {{zb:0451.54017}}, where it is called "PCGS" (path connected generated space). +This property is used in the literature, but usually without a separate name. +One exception is {{zb:0451.54017}} (accessible [here](https://topology.nipissingu.ca/tp/reprints/v04/tp04204.pdf)), +where it is called "PCGS" (path connectedly generated space). Compare with {P42}. From e949c044d9bec745a4d2c62441366a6e06c385d2 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Fri, 13 Mar 2026 00:54:48 -0400 Subject: [PATCH 5/6] Make T108 more like T89 --- theorems/T000108.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/theorems/T000108.md b/theorems/T000108.md index 990e0d7f7..0be0af4c2 100644 --- a/theorems/T000108.md +++ b/theorems/T000108.md @@ -2,10 +2,11 @@ uid: T000108 if: and: - - P000047: true - P000234: true + - P000047: true then: P000052: true --- -If every point has a connected neighborhood and the only connected sets are single points, then every point has a neighborhood consisting of the point itself. +By {P47} the connected components of $X$ are singletons, +so by {P234} every singleton is open. From 86053d90c1e7dd1e0b49190fa1c94fa6d4f80466 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Fri, 13 Mar 2026 01:06:14 -0400 Subject: [PATCH 6/6] "compare with" --- properties/P000233.md | 2 +- properties/P000234.md | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/properties/P000233.md b/properties/P000233.md index 36cf4a2ce..c3f895933 100644 --- a/properties/P000233.md +++ b/properties/P000233.md @@ -24,7 +24,7 @@ This property is used in the literature, but usually without a separate name. One exception is {{zb:0451.54017}} (accessible [here](https://topology.nipissingu.ca/tp/reprints/v04/tp04204.pdf)), where it is called "PCGS" (path connectedly generated space). -Compare with {P42}. +Compare with {P42} and {P234}. ---- #### Meta-properties diff --git a/properties/P000234.md b/properties/P000234.md index 63b7e7a25..885698774 100644 --- a/properties/P000234.md +++ b/properties/P000234.md @@ -18,7 +18,7 @@ This is equivalent to each of the following: The equivalence between the various conditions can be shown by arguments similar to those used in the definition of {P42}, in particular {{mathse:3002235}}. -Compare with {P41}. +Compare with {P41} and {P233}. ---- #### Meta-properties