From 26df1819c01621160ee0a82297259be684e4738d Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 19:19:33 -0500 Subject: [PATCH 01/35] adds P230 --- properties/P000230.md | 17 +++++++++++++++++ 1 file changed, 17 insertions(+) create mode 100644 properties/P000230.md diff --git a/properties/P000230.md b/properties/P000230.md new file mode 100644 index 000000000..c175a8d3a --- /dev/null +++ b/properties/P000230.md @@ -0,0 +1,17 @@ +--- +uid: P000230 +name: Locally simply connected +refs: + - wikipedia: Locally simply connected space + name: Locally simply connected space on Wikipedia + - mr: 2766102 + name: Introduction to topological manifolds (Lee) + - zb: "0951.54001" + name: Topology (Munkres) + - mo: 487326 + name: 'Definition of locally simply connected space' +--- + +$X$ admits a basis of simply connected open sets. + +Defined on page 298 of {{mr:2766102}} and on page 495 of {{zb:0951.54001}}. From 7a029157c9ecf93d04468a2f15ea6b0c5cb1db8b Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 19:27:32 -0500 Subject: [PATCH 02/35] Adds weakly locally simply connected and mentions MO thread in both files now. --- properties/P000230.md | 2 +- properties/P000231.md | 13 +++++++++++++ 2 files changed, 14 insertions(+), 1 deletion(-) create mode 100644 properties/P000231.md diff --git a/properties/P000230.md b/properties/P000230.md index c175a8d3a..cbf9b1431 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -14,4 +14,4 @@ refs: $X$ admits a basis of simply connected open sets. -Defined on page 298 of {{mr:2766102}} and on page 495 of {{zb:0951.54001}}. +Defined on page 298 of {{mr:2766102}} and on page 495 of {{zb:0951.54001}}. Property $P_1$ of {{mo:487326}}. diff --git a/properties/P000231.md b/properties/P000231.md new file mode 100644 index 000000000..41d29b40f --- /dev/null +++ b/properties/P000231.md @@ -0,0 +1,13 @@ +--- +uid: P000231 +name: Weakly locally simply connected +refs: + - zb: "0063.00842" + name: Theory of Lie groups. I (Chevalley) + - mo: 487326 + name: 'Definition of locally simply connected space' +--- + +$X$ admits a basis of simply connected open sets. + +Defined on page 54 of {{zb:0063.00842}}. Property $P_4$ of {{mo:487326}}. From 0f7c6a2b266a704d08e5c6ae7e0c95257cbb7b49 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 19:30:02 -0500 Subject: [PATCH 03/35] Adding definition to P231 --- properties/P000231.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000231.md b/properties/P000231.md index 41d29b40f..6348813c8 100644 --- a/properties/P000231.md +++ b/properties/P000231.md @@ -8,6 +8,6 @@ refs: name: 'Definition of locally simply connected space' --- -$X$ admits a basis of simply connected open sets. +Every point of $X$ has at least one simply connected neighborhood. Defined on page 54 of {{zb:0063.00842}}. Property $P_4$ of {{mo:487326}}. From 16e0009b376af96eee83f2a2b5baa1e34bf1ad80 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 19:35:26 -0500 Subject: [PATCH 04/35] Use P200 in definition and follow suggestion about mentioning 'locally simply connected' in body. --- properties/P000231.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/properties/P000231.md b/properties/P000231.md index 6348813c8..bb6100d09 100644 --- a/properties/P000231.md +++ b/properties/P000231.md @@ -8,6 +8,6 @@ refs: name: 'Definition of locally simply connected space' --- -Every point of $X$ has at least one simply connected neighborhood. +Every point of $X$ has a neighborhood which is P000200. -Defined on page 54 of {{zb:0063.00842}}. Property $P_4$ of {{mo:487326}}. +Defined as "locally simply connected" on page 54 of {{zb:0063.00842}}. Property $P_4$ of {{mo:487326}}. From c5fb9cc4d1eb26ddf63334e056e7d3b06c946caf Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 19:36:53 -0500 Subject: [PATCH 05/35] I forgot []'s --- properties/P000231.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000231.md b/properties/P000231.md index bb6100d09..0647c97bd 100644 --- a/properties/P000231.md +++ b/properties/P000231.md @@ -8,6 +8,6 @@ refs: name: 'Definition of locally simply connected space' --- -Every point of $X$ has a neighborhood which is P000200. +Every point of $X$ has a neighborhood which is [P000200]. Defined as "locally simply connected" on page 54 of {{zb:0063.00842}}. Property $P_4$ of {{mo:487326}}. From 966305408c2984f412656684a3bdcaaf54dac235 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 19:38:27 -0500 Subject: [PATCH 06/35] I forgot it's {}'s braces, and we remove the extra 0's. --- properties/P000231.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000231.md b/properties/P000231.md index 0647c97bd..e2f49ca90 100644 --- a/properties/P000231.md +++ b/properties/P000231.md @@ -8,6 +8,6 @@ refs: name: 'Definition of locally simply connected space' --- -Every point of $X$ has a neighborhood which is [P000200]. +Every point of $X$ has a neighborhood which is {P200}. Defined as "locally simply connected" on page 54 of {{zb:0063.00842}}. Property $P_4$ of {{mo:487326}}. From 6341749db11a7eaf3516e4e00e793047eba931a3 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 19:40:04 -0500 Subject: [PATCH 07/35] Do the same for P230 --- properties/P000230.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000230.md b/properties/P000230.md index cbf9b1431..2a2d27d1f 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -12,6 +12,6 @@ refs: name: 'Definition of locally simply connected space' --- -$X$ admits a basis of simply connected open sets. +$X$ admits a basis of open sets which are {P200}. Defined on page 298 of {{mr:2766102}} and on page 495 of {{zb:0951.54001}}. Property $P_1$ of {{mo:487326}}. From 219dc2d25881c70d36c97ff44ee2add6eb5b5ac8 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 20:23:06 -0500 Subject: [PATCH 08/35] Adds LC^1 --- properties/P000232.md | 25 +++++++++++++++++++++++++ 1 file changed, 25 insertions(+) create mode 100644 properties/P000232.md diff --git a/properties/P000232.md b/properties/P000232.md new file mode 100644 index 000000000..2f326d9cf --- /dev/null +++ b/properties/P000232.md @@ -0,0 +1,25 @@ +--- +uid: P000232 +name: $LC^1$ +alias: + - Locally simply connected +refs: + - zb: "0153.52905" + name: Theory of retracts (Borsuk) + - mo: 487326 + name: 'Definition of locally simply connected space' + - mathse: 5126526 + name: Is Borsuk's definition of $LC^1$ equivalent to this formulation of 'locally simply connected' ($P_{10}$)? +--- + +$X$ is locally $0$-connected and locally $1$-connected, as in {{zb:0153.52905}}. +A space $X$ is locally $n$-connected if for every $x \in X$ and each neighborhood $N$ of $x$ in $X$, there is a neighborhood $U$ of $x$ contained in $N$ such that every map $S^n \to N$ with values in $U$ is null-homotopic in $N$. + +Equivalently, for every $x \in X$ and each neighborhood $N$ of $x$ in $X$, there is a path-connected neighborhood $U$ of $x$ contained in $N$ such that every loop in $U$ is null-homotopic in $N$. See {{mathse:5126526}}. + +This property is part of a hierarchy of $LC^n$ properties which includes {P42} as $LC^0$. +Defined on page 54 of {{zb:0153.52905}}. Property $P_{10}$ of {{mo:487326}}. + +---- +#### Meta-properties +- This property is preserved by retractions. From 4da0912b78f3952c972b2e5237da70d14f04e83c Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 20:28:44 -0500 Subject: [PATCH 09/35] change 'alias' to 'aliases' to fix compile error --- properties/P000232.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000232.md b/properties/P000232.md index 2f326d9cf..612a29446 100644 --- a/properties/P000232.md +++ b/properties/P000232.md @@ -1,7 +1,7 @@ --- uid: P000232 name: $LC^1$ -alias: +aliases: - Locally simply connected refs: - zb: "0153.52905" From db4c6ddb3f6b1a4b95fe7a33f0ab82a24efe445e Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 20:31:52 -0500 Subject: [PATCH 10/35] P_1 implies P_4 --- theorems/T000853.md | 9 +++++++++ 1 file changed, 9 insertions(+) create mode 100644 theorems/T000853.md diff --git a/theorems/T000853.md b/theorems/T000853.md new file mode 100644 index 000000000..e6b957eec --- /dev/null +++ b/theorems/T000853.md @@ -0,0 +1,9 @@ +--- +uid: T000853 +if: + P000230: true +then: + P000231: true +--- + +Immediate from the definitions. From 4dcea093c82d81fe163926801383557f484ff35c Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 20:38:58 -0500 Subject: [PATCH 11/35] P_1 implies P_10 --- properties/P000230.md | 2 ++ theorems/T000854.md | 9 +++++++++ 2 files changed, 11 insertions(+) create mode 100644 theorems/T000854.md diff --git a/properties/P000230.md b/properties/P000230.md index 2a2d27d1f..9a771861a 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -14,4 +14,6 @@ refs: $X$ admits a basis of open sets which are {P200}. +Equivalently, for every $x \in X$ and each neighborhood $N$ of $x$ in $X$, there is a simply connected open neighborhood $U$ of $x$ contained in $N$. + Defined on page 298 of {{mr:2766102}} and on page 495 of {{zb:0951.54001}}. Property $P_1$ of {{mo:487326}}. diff --git a/theorems/T000854.md b/theorems/T000854.md new file mode 100644 index 000000000..f7b145bcf --- /dev/null +++ b/theorems/T000854.md @@ -0,0 +1,9 @@ +--- +uid: T000854 +if: + P000230: true +then: + P000232: true +--- + +Immediate from the equivalent characterizations of each property. From cf4b729f20691278c71a6e3e22b075e23296b9eb Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Wed, 4 Mar 2026 20:41:59 -0500 Subject: [PATCH 12/35] P_4 implies SLSC --- theorems/T000855.md | 9 +++++++++ 1 file changed, 9 insertions(+) create mode 100644 theorems/T000855.md diff --git a/theorems/T000855.md b/theorems/T000855.md new file mode 100644 index 000000000..8dbd02379 --- /dev/null +++ b/theorems/T000855.md @@ -0,0 +1,9 @@ +--- +uid: T000855 +if: + P000231: true +then: + P000229: true +--- + +Immediate from the definitions. From 11987d6db209be6d970017cf41b9bd657cf89dd7 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 01:28:16 -0500 Subject: [PATCH 13/35] LC^1 implies SLSC --- theorems/T000856.md | 13 +++++++++++++ 1 file changed, 13 insertions(+) create mode 100644 theorems/T000856.md diff --git a/theorems/T000856.md b/theorems/T000856.md new file mode 100644 index 000000000..ca0af79a5 --- /dev/null +++ b/theorems/T000856.md @@ -0,0 +1,13 @@ +--- +uid: T000856 +if: + P000232: true +then: + P000229: true +refs: +- mathse: 4044399 + name: Characterizing simply connected spaces +--- + +Let $x \in X$. By {P232}, there exists a path-connected neighborhood $U$ of $x$ such that every loop $ S^1 \to X$ be a loop with image in $U$ is null-homotopic. Let $\sigma$ be a loop in $U$ based at $x$. Then exists a null-homotopy $F : S^1 \times [0, 1] \to X$ of $\sigma$ to a constant loop. Applying arguments $(4) \Rightarrow (5)$ and $(5) \Rightarrow (1)$ from {mathse:4044399} to $F$ constructs a basepoint preserving null-homotopy of $\sigma$ to the constant loop at $x$. +Therefore {P229} follows. From b4bc5b6b919747505128b5a214b91170348b8989 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 01:31:32 -0500 Subject: [PATCH 14/35] Upgrade T847 from SLSC to locally sc --- theorems/T000847.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000847.md b/theorems/T000847.md index 1c68ff60f..5a3261a9e 100644 --- a/theorems/T000847.md +++ b/theorems/T000847.md @@ -3,7 +3,7 @@ uid: T000847 if: P000122: true then: - P000229: true + P000230: true refs: - zb: "0951.54001" name: Topology (Munkres) From deb849067a1dc4d331e027dd0cfe17cdaf9e3e05 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 01:37:06 -0500 Subject: [PATCH 15/35] Fixed typesetting issue --- theorems/T000856.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/theorems/T000856.md b/theorems/T000856.md index ca0af79a5..f335c89f4 100644 --- a/theorems/T000856.md +++ b/theorems/T000856.md @@ -9,5 +9,4 @@ refs: name: Characterizing simply connected spaces --- -Let $x \in X$. By {P232}, there exists a path-connected neighborhood $U$ of $x$ such that every loop $ S^1 \to X$ be a loop with image in $U$ is null-homotopic. Let $\sigma$ be a loop in $U$ based at $x$. Then exists a null-homotopy $F : S^1 \times [0, 1] \to X$ of $\sigma$ to a constant loop. Applying arguments $(4) \Rightarrow (5)$ and $(5) \Rightarrow (1)$ from {mathse:4044399} to $F$ constructs a basepoint preserving null-homotopy of $\sigma$ to the constant loop at $x$. -Therefore {P229} follows. +Let $x \in X$. By {P232}, there exists a path-connected neighborhood $U$ of $x$ such that every loop $ S^1 \to X$ be a loop with image in $U$ is null-homotopic. Let $\sigma$ be a loop in $U$ based at $x$. Then exists a null-homotopy $F : S^1 \times [0, 1] \to X$ of $\sigma$ to a constant loop. Apply the arguments $(4) \Rightarrow (5)$ and $(5) \Rightarrow (1)$ from {{mathse:4044399}} to $F$ in order to construct a basepoint preserving null-homotopy of $\sigma$ to the constant loop at $x$. Then {P229} follows. From 6842f4ea8675b2c17cc930241b548c8c4b833084 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 01:42:33 -0500 Subject: [PATCH 16/35] I accidentally proved weakly locally simply connected previously --- theorems/T000847.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/theorems/T000847.md b/theorems/T000847.md index 5a3261a9e..a43f9e0fc 100644 --- a/theorems/T000847.md +++ b/theorems/T000847.md @@ -9,5 +9,5 @@ refs: name: Topology (Munkres) --- -For $x \in X$ pick a neighborhood $U$ homeomorphic to $\mathbb{R}^n$. -Then $\pi_1(U,x)$ is trivial (see Example 1 on page 331 of {{zb:0951.54001}}). +A locally Euclidean space admits a basis of Euclidean open balls. +For a Euclidean open ball $U$ and $x \in U$, $\pi_1(U,x)$ is trivial (see Example 1 on page 331 of {{zb:0951.54001}}). From efd3899c9a441b128a770ee05489da13514f049b Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 01:44:27 -0500 Subject: [PATCH 17/35] Add newlines for legibility in terminal --- theorems/T000856.md | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/theorems/T000856.md b/theorems/T000856.md index f335c89f4..52fd02b24 100644 --- a/theorems/T000856.md +++ b/theorems/T000856.md @@ -9,4 +9,9 @@ refs: name: Characterizing simply connected spaces --- -Let $x \in X$. By {P232}, there exists a path-connected neighborhood $U$ of $x$ such that every loop $ S^1 \to X$ be a loop with image in $U$ is null-homotopic. Let $\sigma$ be a loop in $U$ based at $x$. Then exists a null-homotopy $F : S^1 \times [0, 1] \to X$ of $\sigma$ to a constant loop. Apply the arguments $(4) \Rightarrow (5)$ and $(5) \Rightarrow (1)$ from {{mathse:4044399}} to $F$ in order to construct a basepoint preserving null-homotopy of $\sigma$ to the constant loop at $x$. Then {P229} follows. +Let $x \in X$. By {P232}, there exists a path-connected neighborhood $U$ of $x$ such that every +loop $ S^1 \to X$ loop with image in $U$ is null-homotopic. Let $\sigma$ be a loop in $U$ based at $x$. +Then exists a null-homotopy $F : S^1 \times [0, 1] \to X$ of $\sigma$ to a constant loop. Apply the arguments +$(4) \Rightarrow (5)$ and $(5) \Rightarrow (1)$ from {{mathse:4044399}} to $F$ in order to construct a +basepoint preserving null-homotopy of $\sigma$ to the constant loop at $x$. +Then {P229} follows. From 093cd043092f4dc299c74f0376d27422af5db667 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 01:45:06 -0500 Subject: [PATCH 18/35] typo --- theorems/T000856.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000856.md b/theorems/T000856.md index 52fd02b24..5869cc615 100644 --- a/theorems/T000856.md +++ b/theorems/T000856.md @@ -11,7 +11,7 @@ refs: Let $x \in X$. By {P232}, there exists a path-connected neighborhood $U$ of $x$ such that every loop $ S^1 \to X$ loop with image in $U$ is null-homotopic. Let $\sigma$ be a loop in $U$ based at $x$. -Then exists a null-homotopy $F : S^1 \times [0, 1] \to X$ of $\sigma$ to a constant loop. Apply the arguments +Choose a null-homotopy $F : S^1 \times [0, 1] \to X$ of $\sigma$ to a constant loop. Apply the arguments $(4) \Rightarrow (5)$ and $(5) \Rightarrow (1)$ from {{mathse:4044399}} to $F$ in order to construct a basepoint preserving null-homotopy of $\sigma$ to the constant loop at $x$. Then {P229} follows. From 7a7dbf69ca4b00255392f780c2bd2309dee3e63e Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 01:46:15 -0500 Subject: [PATCH 19/35] change word --- theorems/T000856.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000856.md b/theorems/T000856.md index 5869cc615..d681b476c 100644 --- a/theorems/T000856.md +++ b/theorems/T000856.md @@ -11,7 +11,7 @@ refs: Let $x \in X$. By {P232}, there exists a path-connected neighborhood $U$ of $x$ such that every loop $ S^1 \to X$ loop with image in $U$ is null-homotopic. Let $\sigma$ be a loop in $U$ based at $x$. -Choose a null-homotopy $F : S^1 \times [0, 1] \to X$ of $\sigma$ to a constant loop. Apply the arguments +Choose a null-homotopy $F : S^1 \times [0, 1] \to X$ from $\sigma$ to a constant loop. Apply the arguments $(4) \Rightarrow (5)$ and $(5) \Rightarrow (1)$ from {{mathse:4044399}} to $F$ in order to construct a basepoint preserving null-homotopy of $\sigma$ to the constant loop at $x$. Then {P229} follows. From cac8d3acd922b80fbb15055692f41149b13bcc6a Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 01:49:36 -0500 Subject: [PATCH 20/35] LC^1 implies locally path connected --- theorems/T000857.md | 9 +++++++++ 1 file changed, 9 insertions(+) create mode 100644 theorems/T000857.md diff --git a/theorems/T000857.md b/theorems/T000857.md new file mode 100644 index 000000000..a3df3fa5c --- /dev/null +++ b/theorems/T000857.md @@ -0,0 +1,9 @@ +--- +uid: T000857 +if: + P000232: true +then: + P000042: true +--- + +$LC^1$ implies $LC^0$ by definition. From 42d45900b4fcb11795d858c5ee5d7c0384989e33 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 01:50:56 -0500 Subject: [PATCH 21/35] Alexandrov implies locally simply connected --- theorems/T000848.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000848.md b/theorems/T000848.md index 93c4720bb..3ef87964f 100644 --- a/theorems/T000848.md +++ b/theorems/T000848.md @@ -3,7 +3,7 @@ uid: T000848 if: P000090: true then: - P000229: true + P000230: true refs: - mathse: 2965374 name: Answer to "Are minimal neighborhoods in an Alexandrov topology path-connected?" From 36fb101c3b3436e08ff99fd670654770d2e78666 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 08:15:43 -0500 Subject: [PATCH 22/35] Update properties/P000230.md Co-authored-by: Felix Pernegger --- properties/P000230.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000230.md b/properties/P000230.md index 9a771861a..cf8b3e347 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -16,4 +16,4 @@ $X$ admits a basis of open sets which are {P200}. Equivalently, for every $x \in X$ and each neighborhood $N$ of $x$ in $X$, there is a simply connected open neighborhood $U$ of $x$ contained in $N$. -Defined on page 298 of {{mr:2766102}} and on page 495 of {{zb:0951.54001}}. Property $P_1$ of {{mo:487326}}. +Defined on page 298 of {{mr:2766102}}, page 495 of {{zb:0951.54001}} and as property $P_1$ of {{mo:487326}}. From 084095576044f0a2a7c2b26c8c8aa395e0fae68b Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 08:16:08 -0500 Subject: [PATCH 23/35] Update properties/P000231.md Co-authored-by: Felix Pernegger --- properties/P000231.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000231.md b/properties/P000231.md index e2f49ca90..2c7f69a5f 100644 --- a/properties/P000231.md +++ b/properties/P000231.md @@ -10,4 +10,4 @@ refs: Every point of $X$ has a neighborhood which is {P200}. -Defined as "locally simply connected" on page 54 of {{zb:0063.00842}}. Property $P_4$ of {{mo:487326}}. +Defined as "locally simply connected" on page 54 of {{zb:0063.00842}} and as property $P_4$ of {{mo:487326}}. From 5788ad4825003bba6aeff93d249f04606667b0b7 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 5 Mar 2026 08:16:21 -0500 Subject: [PATCH 24/35] Update properties/P000232.md Co-authored-by: Felix Pernegger --- properties/P000232.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000232.md b/properties/P000232.md index 612a29446..0959851d5 100644 --- a/properties/P000232.md +++ b/properties/P000232.md @@ -18,7 +18,7 @@ A space $X$ is locally $n$-connected if for every $x \in X$ and each neighborhoo Equivalently, for every $x \in X$ and each neighborhood $N$ of $x$ in $X$, there is a path-connected neighborhood $U$ of $x$ contained in $N$ such that every loop in $U$ is null-homotopic in $N$. See {{mathse:5126526}}. This property is part of a hierarchy of $LC^n$ properties which includes {P42} as $LC^0$. -Defined on page 54 of {{zb:0153.52905}}. Property $P_{10}$ of {{mo:487326}}. +Defined on page 54 of {{zb:0153.52905}} and as property $P_{10}$ of {{mo:487326}}. ---- #### Meta-properties From b757ca8365d1b84b7e215da3b7e242181b7a60b4 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 19:37:56 -0500 Subject: [PATCH 25/35] Address felixpernegger's comment about euclidean ball path connected --- theorems/T000847.md | 1 + 1 file changed, 1 insertion(+) diff --git a/theorems/T000847.md b/theorems/T000847.md index a43f9e0fc..131e5ba0e 100644 --- a/theorems/T000847.md +++ b/theorems/T000847.md @@ -11,3 +11,4 @@ refs: A locally Euclidean space admits a basis of Euclidean open balls. For a Euclidean open ball $U$ and $x \in U$, $\pi_1(U,x)$ is trivial (see Example 1 on page 331 of {{zb:0951.54001}}). +The claim follows because $\mathbb{R}^n$ is contractible, and {T583}. From 8d2010f25baea1d6f5cdb61c0b0457f3bd9e8e6c Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 20:06:32 -0500 Subject: [PATCH 26/35] Remove wikipedia and Munkres from P230 --- properties/P000230.md | 4 ---- 1 file changed, 4 deletions(-) diff --git a/properties/P000230.md b/properties/P000230.md index cf8b3e347..03db5873a 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -2,12 +2,8 @@ uid: P000230 name: Locally simply connected refs: - - wikipedia: Locally simply connected space - name: Locally simply connected space on Wikipedia - mr: 2766102 name: Introduction to topological manifolds (Lee) - - zb: "0951.54001" - name: Topology (Munkres) - mo: 487326 name: 'Definition of locally simply connected space' --- From 353ed519d7b249c2ec52dd303712981e159c99b5 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 20:07:31 -0500 Subject: [PATCH 27/35] Update properties/P000230.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000230.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000230.md b/properties/P000230.md index 03db5873a..39734e7b6 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -10,6 +10,6 @@ refs: $X$ admits a basis of open sets which are {P200}. -Equivalently, for every $x \in X$ and each neighborhood $N$ of $x$ in $X$, there is a simply connected open neighborhood $U$ of $x$ contained in $N$. +Equivalently, for each $x \in X$ every neighborhood of $x$ contains a simply connected open neighborhood of $x$. Defined on page 298 of {{mr:2766102}}, page 495 of {{zb:0951.54001}} and as property $P_1$ of {{mo:487326}}. From 8f722b415fa96a093df6b5780bfb8142e4196dd8 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 20:09:51 -0500 Subject: [PATCH 28/35] comma --- properties/P000230.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000230.md b/properties/P000230.md index 39734e7b6..41ab083e5 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -10,6 +10,6 @@ refs: $X$ admits a basis of open sets which are {P200}. -Equivalently, for each $x \in X$ every neighborhood of $x$ contains a simply connected open neighborhood of $x$. +Equivalently, for each $x \in X$, every neighborhood of $x$ contains a simply connected open neighborhood of $x$. Defined on page 298 of {{mr:2766102}}, page 495 of {{zb:0951.54001}} and as property $P_1$ of {{mo:487326}}. From 4564d74b0d5566f2149a5ef8b97903d00799e9fb Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 20:10:48 -0500 Subject: [PATCH 29/35] Update properties/P000230.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000230.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000230.md b/properties/P000230.md index 41ab083e5..c9041a7e8 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -2,7 +2,7 @@ uid: P000230 name: Locally simply connected refs: - - mr: 2766102 + - zb: "1209.57001" name: Introduction to topological manifolds (Lee) - mo: 487326 name: 'Definition of locally simply connected space' From 6cd8a0a29bc9573922ac4b20403d350df9a4cf21 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 20:13:55 -0500 Subject: [PATCH 30/35] The commit got outdated, so manually switch in text to zbmath from https://github.com/pi-base/data/pull/1659#discussion_r2894066716 --- properties/P000230.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000230.md b/properties/P000230.md index c9041a7e8..bd7ee437d 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -12,4 +12,4 @@ $X$ admits a basis of open sets which are {P200}. Equivalently, for each $x \in X$, every neighborhood of $x$ contains a simply connected open neighborhood of $x$. -Defined on page 298 of {{mr:2766102}}, page 495 of {{zb:0951.54001}} and as property $P_1$ of {{mo:487326}}. +Defined on page 298 of {{zb:1209.57001}}, page 495 of {{zb:0951.54001}} and as property $P_1$ of {{mo:487326}}. From 271ca85853de9dcc17c6bcfb5cd67bf5ffcc937a Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 20:20:15 -0500 Subject: [PATCH 31/35] Mention strongly locally simply connected inline in P230, but not as an alias. --- properties/P000230.md | 3 +++ 1 file changed, 3 insertions(+) diff --git a/properties/P000230.md b/properties/P000230.md index bd7ee437d..119073fe7 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -4,6 +4,7 @@ name: Locally simply connected refs: - zb: "1209.57001" name: Introduction to topological manifolds (Lee) + - zb: "1030.57035" - mo: 487326 name: 'Definition of locally simply connected space' --- @@ -13,3 +14,5 @@ $X$ admits a basis of open sets which are {P200}. Equivalently, for each $x \in X$, every neighborhood of $x$ contains a simply connected open neighborhood of $x$. Defined on page 298 of {{zb:1209.57001}}, page 495 of {{zb:0951.54001}} and as property $P_1$ of {{mo:487326}}. + +Has also been called "strongly locally simply connected", for example in {{zb:1030.57035}}. From 67c96b19ce2f77ce0c0974f3ea8b313b73c5296e Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 20:22:44 -0500 Subject: [PATCH 32/35] Forgot zb reference name. --- properties/P000230.md | 5 ++--- properties/P000232.md | 2 +- 2 files changed, 3 insertions(+), 4 deletions(-) diff --git a/properties/P000230.md b/properties/P000230.md index 119073fe7..5eefb15ac 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -5,6 +5,7 @@ refs: - zb: "1209.57001" name: Introduction to topological manifolds (Lee) - zb: "1030.57035" + name: 'Bing’s dogbone space is not strongly locally simply connected' - mo: 487326 name: 'Definition of locally simply connected space' --- @@ -13,6 +14,4 @@ $X$ admits a basis of open sets which are {P200}. Equivalently, for each $x \in X$, every neighborhood of $x$ contains a simply connected open neighborhood of $x$. -Defined on page 298 of {{zb:1209.57001}}, page 495 of {{zb:0951.54001}} and as property $P_1$ of {{mo:487326}}. - -Has also been called "strongly locally simply connected", for example in {{zb:1030.57035}}. +Defined on page 298 of {{zb:1209.57001}}, page 495 of {{zb:0951.54001}} and as property $P_1$ of {{mo:487326}}. Has also been called "strongly locally simply connected", for example in {{zb:1030.57035}}. diff --git a/properties/P000232.md b/properties/P000232.md index 0959851d5..517ec87df 100644 --- a/properties/P000232.md +++ b/properties/P000232.md @@ -18,7 +18,7 @@ A space $X$ is locally $n$-connected if for every $x \in X$ and each neighborhoo Equivalently, for every $x \in X$ and each neighborhood $N$ of $x$ in $X$, there is a path-connected neighborhood $U$ of $x$ contained in $N$ such that every loop in $U$ is null-homotopic in $N$. See {{mathse:5126526}}. This property is part of a hierarchy of $LC^n$ properties which includes {P42} as $LC^0$. -Defined on page 54 of {{zb:0153.52905}} and as property $P_{10}$ of {{mo:487326}}. +Defined on page 54 of {{zb:0153.52905}} and as property $P_{10}$ of {{mo:487326}}. Has also been called ---- #### Meta-properties From 59ab195b35bd57022f75e4b14c7fbfa798823459 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 20:25:32 -0500 Subject: [PATCH 33/35] I'm guessing refs should be listed in appearance order. --- properties/P000230.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/properties/P000230.md b/properties/P000230.md index 5eefb15ac..a00e4a5cf 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -4,10 +4,10 @@ name: Locally simply connected refs: - zb: "1209.57001" name: Introduction to topological manifolds (Lee) - - zb: "1030.57035" - name: 'Bing’s dogbone space is not strongly locally simply connected' - mo: 487326 name: 'Definition of locally simply connected space' + - zb: "1030.57035" + name: 'Bing’s dogbone space is not strongly locally simply connected' --- $X$ admits a basis of open sets which are {P200}. From 77d187044d2bb95a7002a359feefd72b43aa58cc Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 20:33:52 -0500 Subject: [PATCH 34/35] Add paper using the term "locally simply connected" for P232. --- properties/P000230.md | 2 +- properties/P000232.md | 4 +++- 2 files changed, 4 insertions(+), 2 deletions(-) diff --git a/properties/P000230.md b/properties/P000230.md index a00e4a5cf..c33916d4d 100644 --- a/properties/P000230.md +++ b/properties/P000230.md @@ -7,7 +7,7 @@ refs: - mo: 487326 name: 'Definition of locally simply connected space' - zb: "1030.57035" - name: 'Bing’s dogbone space is not strongly locally simply connected' + name: Bing's dogbone space is not strongly locally simply connected --- $X$ admits a basis of open sets which are {P200}. diff --git a/properties/P000232.md b/properties/P000232.md index 517ec87df..0af6da2a2 100644 --- a/properties/P000232.md +++ b/properties/P000232.md @@ -10,6 +10,8 @@ refs: name: 'Definition of locally simply connected space' - mathse: 5126526 name: Is Borsuk's definition of $LC^1$ equivalent to this formulation of 'locally simply connected' ($P_{10}$)? + - zb: "0198.56303" + name: Acyclicity in three-manifolds --- $X$ is locally $0$-connected and locally $1$-connected, as in {{zb:0153.52905}}. @@ -18,7 +20,7 @@ A space $X$ is locally $n$-connected if for every $x \in X$ and each neighborhoo Equivalently, for every $x \in X$ and each neighborhood $N$ of $x$ in $X$, there is a path-connected neighborhood $U$ of $x$ contained in $N$ such that every loop in $U$ is null-homotopic in $N$. See {{mathse:5126526}}. This property is part of a hierarchy of $LC^n$ properties which includes {P42} as $LC^0$. -Defined on page 54 of {{zb:0153.52905}} and as property $P_{10}$ of {{mo:487326}}. Has also been called +Defined on page 54 of {{zb:0153.52905}} and as property $P_{10}$ of {{mo:487326}}. Has also been called "locally simply connected", for example in {{zb:0198.56303}}. ---- #### Meta-properties From 1380e5a72e3483b21725f9f7264c578864f36d4a Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 6 Mar 2026 20:41:08 -0500 Subject: [PATCH 35/35] Change second paragraph of P232 on prabau's suggestion. --- properties/P000232.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000232.md b/properties/P000232.md index 0af6da2a2..d3a48dd89 100644 --- a/properties/P000232.md +++ b/properties/P000232.md @@ -17,7 +17,7 @@ refs: $X$ is locally $0$-connected and locally $1$-connected, as in {{zb:0153.52905}}. A space $X$ is locally $n$-connected if for every $x \in X$ and each neighborhood $N$ of $x$ in $X$, there is a neighborhood $U$ of $x$ contained in $N$ such that every map $S^n \to N$ with values in $U$ is null-homotopic in $N$. -Equivalently, for every $x \in X$ and each neighborhood $N$ of $x$ in $X$, there is a path-connected neighborhood $U$ of $x$ contained in $N$ such that every loop in $U$ is null-homotopic in $N$. See {{mathse:5126526}}. +Equivalently, for each $x \in X$, every neighborhood $N$ of $x$ contains a {P37} neighborhood $U$ of $x$ such that every loop $\phi:S^1\to U$ is null-homotopic in $N$. See {{mathse:5126526}}. This property is part of a hierarchy of $LC^n$ properties which includes {P42} as $LC^0$. Defined on page 54 of {{zb:0153.52905}} and as property $P_{10}$ of {{mo:487326}}. Has also been called "locally simply connected", for example in {{zb:0198.56303}}.