From 2f661df9b1ee5de31c2ffdff6ef0494dd0079056 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 25 Apr 2026 20:47:51 +0200 Subject: [PATCH 01/32] cellular --- properties/P000007.md | 1 + properties/P000240.md | 18 ++++++++++++++++++ theorems/T000883.md | 12 ++++++++++++ theorems/T000884.md | 12 ++++++++++++ 4 files changed, 43 insertions(+) create mode 100644 properties/P000240.md create mode 100644 theorems/T000883.md create mode 100644 theorems/T000884.md diff --git a/properties/P000007.md b/properties/P000007.md index 66128adda5..534bf48f7d 100644 --- a/properties/P000007.md +++ b/properties/P000007.md @@ -4,6 +4,7 @@ name: "$T_4$" aliases: - Normal Hausdorff - T4 + - Normal refs: - zb: "1052.54001" name: General Topology (Willard) diff --git a/properties/P000240.md b/properties/P000240.md new file mode 100644 index 0000000000..c8ba2664e8 --- /dev/null +++ b/properties/P000240.md @@ -0,0 +1,18 @@ +--- +uid: P000241 +name: Cellular +aliases: + - Has a CW structure +refs: + - wikipedia: CW complex + name: CW complex on Wikipedia + - zb: "1044.55001" + name: Algebraic Topology (Hatcher) +--- + +$X$ is homeomorphic to the underlying space of a [CW complex](https://en.wikipedia.org/wiki/CW_complex). + +---- +#### Meta-properties + +- This property is preserved by arbitrary disjoint unions. diff --git a/theorems/T000883.md b/theorems/T000883.md new file mode 100644 index 0000000000..5a157e1052 --- /dev/null +++ b/theorems/T000883.md @@ -0,0 +1,12 @@ +--- +uid: T000882 +if: + P000240: true +then: + P000223: true +refs: + - zb: "1044.55001" + name: Algebraic Topology (Hatcher) +--- + +See Proposition A.4 in {{zb:1044.55001}}. diff --git a/theorems/T000884.md b/theorems/T000884.md new file mode 100644 index 0000000000..4eac3e32ea --- /dev/null +++ b/theorems/T000884.md @@ -0,0 +1,12 @@ +--- +uid: T000882 +if: + P000240: true +then: + P000007: true +refs: + - zb: "1044.55001" + name: Algebraic Topology (Hatcher) +--- + +See Proposition A.3 in {{zb:1044.55001}}, where "normal" is defined as {P7}. From 697f6e64d0f36d2bdf220df0885ff7707f2f5e0f Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 25 Apr 2026 20:48:58 +0200 Subject: [PATCH 02/32] fix uid --- theorems/T000883.md | 2 +- theorems/T000884.md | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/theorems/T000883.md b/theorems/T000883.md index 5a157e1052..fe5daf2b15 100644 --- a/theorems/T000883.md +++ b/theorems/T000883.md @@ -1,5 +1,5 @@ --- -uid: T000882 +uid: T000883 if: P000240: true then: diff --git a/theorems/T000884.md b/theorems/T000884.md index 4eac3e32ea..fb463eddfc 100644 --- a/theorems/T000884.md +++ b/theorems/T000884.md @@ -1,5 +1,5 @@ --- -uid: T000882 +uid: T000884 if: P000240: true then: From ff5634e74bedf35175d4d9108c190ba0c70ca1c0 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 25 Apr 2026 20:59:55 +0200 Subject: [PATCH 03/32] fix uid once more --- properties/P000240.md | 2 +- theorems/T000883.md | 4 ++-- theorems/T000884.md | 4 ++-- 3 files changed, 5 insertions(+), 5 deletions(-) diff --git a/properties/P000240.md b/properties/P000240.md index c8ba2664e8..69a1281b40 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -1,5 +1,5 @@ --- -uid: P000241 +uid: P000240 name: Cellular aliases: - Has a CW structure diff --git a/theorems/T000883.md b/theorems/T000883.md index fe5daf2b15..0d11ea8ad7 100644 --- a/theorems/T000883.md +++ b/theorems/T000883.md @@ -3,10 +3,10 @@ uid: T000883 if: P000240: true then: - P000223: true + P000007: true refs: - zb: "1044.55001" name: Algebraic Topology (Hatcher) --- -See Proposition A.4 in {{zb:1044.55001}}. +See Proposition A.3 in {{zb:1044.55001}}, where "normal" is defined as {P7}. diff --git a/theorems/T000884.md b/theorems/T000884.md index fb463eddfc..7ae33cddb0 100644 --- a/theorems/T000884.md +++ b/theorems/T000884.md @@ -3,10 +3,10 @@ uid: T000884 if: P000240: true then: - P000007: true + P000223: true refs: - zb: "1044.55001" name: Algebraic Topology (Hatcher) --- -See Proposition A.3 in {{zb:1044.55001}}, where "normal" is defined as {P7}. +See Proposition A.4 in {{zb:1044.55001}}. From 406026fb4bfa63892f08650178778da8a7335377 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 25 Apr 2026 22:17:07 +0200 Subject: [PATCH 04/32] paracompact --- theorems/T000885.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) create mode 100644 theorems/T000885.md diff --git a/theorems/T000885.md b/theorems/T000885.md new file mode 100644 index 0000000000..0b0d7ccf48 --- /dev/null +++ b/theorems/T000885.md @@ -0,0 +1,12 @@ +--- +uid: T000884 +if: + P000240: true +then: + P000030: true +refs: + - zb: "0837.55001" + name: Cellular structures in topology (Fritsch,Piccinini) +--- + +See Theorem 1.3.5 in {{zb:0837.55001}}. From 38d4baf4e6a3ab52813c582b46b07511a660f6f5 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 25 Apr 2026 22:18:07 +0200 Subject: [PATCH 05/32] fix uid --- theorems/T000885.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000885.md b/theorems/T000885.md index 0b0d7ccf48..ea8a01b983 100644 --- a/theorems/T000885.md +++ b/theorems/T000885.md @@ -1,5 +1,5 @@ --- -uid: T000884 +uid: T000885 if: P000240: true then: From 1ec2ddfcc75660706a0e7fd7ed91266f3f0ebc68 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 26 Apr 2026 02:56:18 +0200 Subject: [PATCH 06/32] compactly generated --- theorems/T000886.md | 9 +++++++++ 1 file changed, 9 insertions(+) create mode 100644 theorems/T000886.md diff --git a/theorems/T000886.md b/theorems/T000886.md new file mode 100644 index 0000000000..b94ebb25ff --- /dev/null +++ b/theorems/T000886.md @@ -0,0 +1,9 @@ +--- +uid: T000886 +if: + P000240: true +then: + P000141: true +--- + +See Proposition 3.3 in . From 8876bcbbf3af61c25213a1f67680eb1204d3c209 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 26 Apr 2026 03:39:14 +0200 Subject: [PATCH 07/32] add def of cw complex --- properties/P000240.md | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/properties/P000240.md b/properties/P000240.md index 69a1281b40..ffdae33157 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -1,8 +1,6 @@ --- uid: P000240 -name: Cellular -aliases: - - Has a CW structure +name: CW complex refs: - wikipedia: CW complex name: CW complex on Wikipedia @@ -10,7 +8,12 @@ refs: name: Algebraic Topology (Hatcher) --- -$X$ is homeomorphic to the underlying space of a [CW complex](https://en.wikipedia.org/wiki/CW_complex). +There exist subspaces $\empty = X_{-1}\subseteq X_0 \subseteq \dots \subseteq X$, such that: + +- $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. +- A subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. + +For an alternative, categorical (and more modern) definition, see . ---- #### Meta-properties From 43a44c3c6b1aa8efd192e015f37b38b95bf01358 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 26 Apr 2026 03:46:52 +0200 Subject: [PATCH 08/32] add disclaimer --- properties/P000240.md | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/properties/P000240.md b/properties/P000240.md index ffdae33157..f565ae53b2 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -1,6 +1,9 @@ --- uid: P000240 name: CW complex +aliases: + - Cellular complex + - Cell complex refs: - wikipedia: CW complex name: CW complex on Wikipedia @@ -13,6 +16,8 @@ There exist subspaces $\empty = X_{-1}\subseteq X_0 \subseteq \dots \subseteq X$ - $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. - A subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. +*Note:* Techincally, a CW complex is a topological space together with such a subset filtration (this is i.e. necessary to define cellular maps). For simplicity, we omit this technicality here. + For an alternative, categorical (and more modern) definition, see . ---- From 013675d111f00929d8eb34a75527aaa8b3350f1d Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 26 Apr 2026 04:08:48 +0200 Subject: [PATCH 09/32] update to t6 --- theorems/T000883.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/theorems/T000883.md b/theorems/T000883.md index 0d11ea8ad7..8c3f62ed8b 100644 --- a/theorems/T000883.md +++ b/theorems/T000883.md @@ -3,10 +3,10 @@ uid: T000883 if: P000240: true then: - P000007: true + P000067: true refs: - - zb: "1044.55001" - name: Algebraic Topology (Hatcher) + - zb: "0207.21704" + name: The Topology of CW Complexes (Lundell, Weingram) --- -See Proposition A.3 in {{zb:1044.55001}}, where "normal" is defined as {P7}. +See Proposition 4.3 in {{zb:0207.21704}}, where "perfectly normal" is defined as {P67}. From 39010132472446c149994c54e1b376e8b0db9fbf Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 26 Apr 2026 04:22:43 +0200 Subject: [PATCH 10/32] add 1 more terms --- properties/P000240.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index f565ae53b2..f529a41f07 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -11,7 +11,7 @@ refs: name: Algebraic Topology (Hatcher) --- -There exist subspaces $\empty = X_{-1}\subseteq X_0 \subseteq \dots \subseteq X$, such that: +$X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots \subseteq X$, such that: - $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. - A subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. From cf456fa1b064e770b16451c8d0001b8e74b6da9c Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 26 Apr 2026 04:28:06 +0200 Subject: [PATCH 11/32] add union --- properties/P000240.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/properties/P000240.md b/properties/P000240.md index f529a41f07..b85c3e28be 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -11,10 +11,10 @@ refs: name: Algebraic Topology (Hatcher) --- -$X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots \subseteq X$, such that: +$X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots$, such that: - $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. -- A subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. +- $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. *Note:* Techincally, a CW complex is a topological space together with such a subset filtration (this is i.e. necessary to define cellular maps). For simplicity, we omit this technicality here. From 11ac1e1e86a294b4cd321e0739ecdf5557a1df4f Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 26 Apr 2026 04:37:31 +0200 Subject: [PATCH 12/32] delete aliases (why not) --- properties/P000240.md | 3 --- 1 file changed, 3 deletions(-) diff --git a/properties/P000240.md b/properties/P000240.md index b85c3e28be..d3ff2c3815 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -1,9 +1,6 @@ --- uid: P000240 name: CW complex -aliases: - - Cellular complex - - Cell complex refs: - wikipedia: CW complex name: CW complex on Wikipedia From a33629cda2e4836063c39b6d11aabbe91c3e0d69 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 26 Apr 2026 04:38:07 +0200 Subject: [PATCH 13/32] typo --- properties/P000240.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index d3ff2c3815..c8e5443df7 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -13,7 +13,7 @@ $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subsete - $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. - $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. -*Note:* Techincally, a CW complex is a topological space together with such a subset filtration (this is i.e. necessary to define cellular maps). For simplicity, we omit this technicality here. +*Note:* Technically, a CW complex is a topological space together with such a subset filtration (this is i.e. necessary to define cellular maps). For simplicity, we omit this technicality here. For an alternative, categorical (and more modern) definition, see . From 9dab4694d0246de904d7c15217025b39290872c2 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 26 Apr 2026 04:54:14 +0200 Subject: [PATCH 14/32] change disclaimer --- properties/P000240.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index c8e5443df7..22e7c0d4d5 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -13,7 +13,8 @@ $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subsete - $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. - $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. -*Note:* Technically, a CW complex is a topological space together with such a subset filtration (this is i.e. necessary to define cellular maps). For simplicity, we omit this technicality here. +*Note:* In standard usage, a CW complex is a topological space together with a chain of subspaces as above (this is i.e. necessary to define cellular maps). +However, we are only concerned with the underlying topological space. For an alternative, categorical (and more modern) definition, see . From 209fbaf091547ffdc1cbe9d45b861d1e625d0648 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 26 Apr 2026 19:04:20 +0200 Subject: [PATCH 15/32] add def attaching cells --- properties/P000240.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/properties/P000240.md b/properties/P000240.md index 22e7c0d4d5..0e7f9f5c00 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -13,6 +13,8 @@ $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subsete - $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. - $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. +Here *attaching (n+1)-cells* means that there is a discrete space $J$ and for every $j \in J$ a continuous map $f_j:S^n \to X_n$ such that $X_{n+1}$ is homeomorph to the quotient $(X_n \sqcup (J \times D^{n+1})) / \sim$, where $\sim$ is the equivalence relation generated by $(j,x)\sim f_j(x)$ for all $x \in J \times S^{n+1}$ (where we identify $\partial D^{n+1}$ with $S^n$). + *Note:* In standard usage, a CW complex is a topological space together with a chain of subspaces as above (this is i.e. necessary to define cellular maps). However, we are only concerned with the underlying topological space. From 5636d3f835a5d202f2f3dfdcaed9e4b726466bf5 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Mon, 27 Apr 2026 05:11:10 +0200 Subject: [PATCH 16/32] Update properties/P000007.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000007.md | 1 - 1 file changed, 1 deletion(-) diff --git a/properties/P000007.md b/properties/P000007.md index 534bf48f7d..66128adda5 100644 --- a/properties/P000007.md +++ b/properties/P000007.md @@ -4,7 +4,6 @@ name: "$T_4$" aliases: - Normal Hausdorff - T4 - - Normal refs: - zb: "1052.54001" name: General Topology (Willard) From 35c1e0faf9feae22a94576856f66a2bbc9b2d270 Mon Sep 17 00:00:00 2001 From: Batixx Date: Mon, 27 Apr 2026 19:47:20 +0200 Subject: [PATCH 17/32] inline S^n --- properties/P000240.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index 0e7f9f5c00..1b41fcd9c0 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -13,7 +13,9 @@ $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subsete - $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. - $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. -Here *attaching (n+1)-cells* means that there is a discrete space $J$ and for every $j \in J$ a continuous map $f_j:S^n \to X_n$ such that $X_{n+1}$ is homeomorph to the quotient $(X_n \sqcup (J \times D^{n+1})) / \sim$, where $\sim$ is the equivalence relation generated by $(j,x)\sim f_j(x)$ for all $x \in J \times S^{n+1}$ (where we identify $\partial D^{n+1}$ with $S^n$). +Here *attaching (n+1)-cells* means that there is a discrete space $J$ and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$ such that $X_{n+1}$ is homeomorph to the quotient $(X_n \sqcup (J \times D^{n+1})) / \sim$, where $\sim$ is the equivalence relation generated by $(j,x)\sim f_j(x)$ for all $x \in J \times \partial D^{n+1}$. + +Here $D^n$ is the closed disk in $n$-dimensions and $\partial D^0 := \emptyset$. *Note:* In standard usage, a CW complex is a topological space together with a chain of subspaces as above (this is i.e. necessary to define cellular maps). However, we are only concerned with the underlying topological space. From f2206e40b34606777db255e7fca85d2b9ec74cfd Mon Sep 17 00:00:00 2001 From: Batixx Date: Mon, 27 Apr 2026 20:42:28 +0200 Subject: [PATCH 18/32] fix ref --- properties/P000240.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index 1b41fcd9c0..af8dc45e7e 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -2,7 +2,7 @@ uid: P000240 name: CW complex refs: - - wikipedia: CW complex + - wikipedia: CW_complex name: CW complex on Wikipedia - zb: "1044.55001" name: Algebraic Topology (Hatcher) From e420dafbf3e49b00f3e4e4a1815abd3c2f8a2777 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Mon, 27 Apr 2026 21:55:24 +0200 Subject: [PATCH 19/32] Update properties/P000240.md Co-authored-by: Geoffrey Sangston --- properties/P000240.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index af8dc45e7e..00e1f4d871 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -13,7 +13,7 @@ $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subsete - $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. - $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. -Here *attaching (n+1)-cells* means that there is a discrete space $J$ and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$ such that $X_{n+1}$ is homeomorph to the quotient $(X_n \sqcup (J \times D^{n+1})) / \sim$, where $\sim$ is the equivalence relation generated by $(j,x)\sim f_j(x)$ for all $x \in J \times \partial D^{n+1}$. +Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) / \sim$, where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $x \in J \times \partial D^{n+1}$. Here $D^n$ is the closed disk in $n$-dimensions and $\partial D^0 := \emptyset$. From 1387c6c070edcf5a37f8f3ec113234f979d5ae6f Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Mon, 27 Apr 2026 22:05:27 +0200 Subject: [PATCH 20/32] remove "modern" --- properties/P000240.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index 00e1f4d871..837204a49b 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -20,7 +20,7 @@ Here $D^n$ is the closed disk in $n$-dimensions and $\partial D^0 := \emptyset$. *Note:* In standard usage, a CW complex is a topological space together with a chain of subspaces as above (this is i.e. necessary to define cellular maps). However, we are only concerned with the underlying topological space. -For an alternative, categorical (and more modern) definition, see . +For an alternative, categorical definition, see . ---- #### Meta-properties From 77fa8d3d0f0a7bac21f3f149220c7c2433915e2a Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Mon, 27 Apr 2026 22:33:42 +0200 Subject: [PATCH 21/32] Update properties/P000240.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000240.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index 837204a49b..04479b661e 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -15,7 +15,7 @@ $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subsete Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) / \sim$, where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $x \in J \times \partial D^{n+1}$. -Here $D^n$ is the closed disk in $n$-dimensions and $\partial D^0 := \emptyset$. +Here $D^n$ is the closed unit disk in $\mathbb R^n$ with its "boundary" $\partial D^n=S^{n-1}$ being the unit sphere in $\mathbb R^n$. In particular, $D^0=\{0\}$ and $\partial D^0=S^{-1}=\emptyset$ by convention. *Note:* In standard usage, a CW complex is a topological space together with a chain of subspaces as above (this is i.e. necessary to define cellular maps). However, we are only concerned with the underlying topological space. From 8c698f71bbfa94259a486d7d545cd1631bb0a529 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 28 Apr 2026 05:55:16 +0200 Subject: [PATCH 22/32] Update properties/P000240.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000240.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index 04479b661e..d968c43e00 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -13,7 +13,8 @@ $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subsete - $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. - $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. -Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) / \sim$, where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $x \in J \times \partial D^{n+1}$. +Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) /{\sim}$, +where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^{n+1}$. Here $D^n$ is the closed unit disk in $\mathbb R^n$ with its "boundary" $\partial D^n=S^{n-1}$ being the unit sphere in $\mathbb R^n$. In particular, $D^0=\{0\}$ and $\partial D^0=S^{-1}=\emptyset$ by convention. From abc3616c984c87261892e84a70c428272c526e4c Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 28 Apr 2026 19:12:00 +0200 Subject: [PATCH 23/32] Update properties/P000240.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000240.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/properties/P000240.md b/properties/P000240.md index d968c43e00..4a229ca9c9 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -15,6 +15,8 @@ $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subsete Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) /{\sim}$, where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^{n+1}$. +The set $J$ is allowed to be empty, in which case $X_{n+1}=X_n$. +The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is empty. Here $D^n$ is the closed unit disk in $\mathbb R^n$ with its "boundary" $\partial D^n=S^{n-1}$ being the unit sphere in $\mathbb R^n$. In particular, $D^0=\{0\}$ and $\partial D^0=S^{-1}=\emptyset$ by convention. From 29943dd6538566336d9d324425d764837f77d634 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Wed, 29 Apr 2026 04:51:28 +0200 Subject: [PATCH 24/32] Apply suggestions from code review Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000240.md | 5 +++-- theorems/T000883.md | 2 +- 2 files changed, 4 insertions(+), 3 deletions(-) diff --git a/properties/P000240.md b/properties/P000240.md index 4a229ca9c9..6f0b5b4bee 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -20,8 +20,9 @@ The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is Here $D^n$ is the closed unit disk in $\mathbb R^n$ with its "boundary" $\partial D^n=S^{n-1}$ being the unit sphere in $\mathbb R^n$. In particular, $D^0=\{0\}$ and $\partial D^0=S^{-1}=\emptyset$ by convention. -*Note:* In standard usage, a CW complex is a topological space together with a chain of subspaces as above (this is i.e. necessary to define cellular maps). -However, we are only concerned with the underlying topological space. +*Note*: A *CW-structure* on a topological space $X$ is a filtration $X_{-1}\subseteq X_0\subseteq X_1 \subseteq\dots$ satisfying the conditions above. +Strictly speaking, a *CW complex* is a space $X$ together with a compatible CW-structure. +For simplicity and by a slight abuse of language, we call the space itself a CW complex if it admits a CW structure. For an alternative, categorical definition, see . diff --git a/theorems/T000883.md b/theorems/T000883.md index 8c3f62ed8b..0823a04e53 100644 --- a/theorems/T000883.md +++ b/theorems/T000883.md @@ -9,4 +9,4 @@ refs: name: The Topology of CW Complexes (Lundell, Weingram) --- -See Proposition 4.3 in {{zb:0207.21704}}, where "perfectly normal" is defined as {P67}. +See Proposition I.4.3 in {{zb:0207.21704}}, where "perfectly normal" is defined as {P67}. From 338731d445344d516abfc8ecf56995256eac9c5f Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Wed, 29 Apr 2026 15:29:15 +0200 Subject: [PATCH 25/32] Update theorems/T000883.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000883.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000883.md b/theorems/T000883.md index 0823a04e53..91d6f0c1f2 100644 --- a/theorems/T000883.md +++ b/theorems/T000883.md @@ -9,4 +9,4 @@ refs: name: The Topology of CW Complexes (Lundell, Weingram) --- -See Proposition I.4.3 in {{zb:0207.21704}}, where "perfectly normal" is defined as {P67}. +See Proposition II.4.3 in {{zb:0207.21704}}, where "perfectly normal" is defined as {P67}. From 0ae65a2bd75ce6ef8768496bd1b3c2b05daa31f8 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Wed, 29 Apr 2026 15:29:33 +0200 Subject: [PATCH 26/32] Update properties/P000240.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000240.md | 60 ++++++++++++++++++++++--------------------- 1 file changed, 31 insertions(+), 29 deletions(-) diff --git a/properties/P000240.md b/properties/P000240.md index 6f0b5b4bee..d3a510b8d0 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -1,32 +1,34 @@ ---- -uid: P000240 -name: CW complex -refs: - - wikipedia: CW_complex - name: CW complex on Wikipedia - - zb: "1044.55001" - name: Algebraic Topology (Hatcher) ---- - -$X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots$, such that: - -- $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. -- $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. - -Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) /{\sim}$, -where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^{n+1}$. -The set $J$ is allowed to be empty, in which case $X_{n+1}=X_n$. -The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is empty. - -Here $D^n$ is the closed unit disk in $\mathbb R^n$ with its "boundary" $\partial D^n=S^{n-1}$ being the unit sphere in $\mathbb R^n$. In particular, $D^0=\{0\}$ and $\partial D^0=S^{-1}=\emptyset$ by convention. - +--- +uid: P000240 +name: CW complex +refs: + - wikipedia: CW_complex + name: CW complex on Wikipedia + - zb: "1044.55001" + name: Algebraic Topology (Hatcher) +--- + +$X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots$, such that: + +- $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. +- $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. + +Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) /{\sim}$, +where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^{n+1}$. +The set $J$ is allowed to be empty, in which case $X_{n+1}=X_n$. +The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is empty. + +Here $D^n$ is the closed unit disk in $\mathbb R^n$ with its "boundary" $\partial D^n=S^{n-1}$ being the unit sphere in $\mathbb R^n$. In particular, $D^0=\{0\}$ and $\partial D^0=S^{-1}=\emptyset$ by convention. + *Note*: A *CW-structure* on a topological space $X$ is a filtration $X_{-1}\subseteq X_0\subseteq X_1 \subseteq\dots$ satisfying the conditions above. Strictly speaking, a *CW complex* is a space $X$ together with a compatible CW-structure. For simplicity and by a slight abuse of language, we call the space itself a CW complex if it admits a CW structure. - -For an alternative, categorical definition, see . - ----- -#### Meta-properties - -- This property is preserved by arbitrary disjoint unions. + +An equivalent definition can be given in terms of a decomposition of $X$ into cells and conditions on their associated characteristic maps. +For details and the meaning of this terminology, see Proposition A.2 on page 521 of {{zb:1044.55001}} or Definition II.1.1 in {{zb:0207.21704}}. +For an alternative, categorical definition, see . + +---- +#### Meta-properties + +- This property is preserved by arbitrary disjoint unions. From d1c4d2714799927675f486e62c76855b85b4ce89 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Wed, 29 Apr 2026 16:08:29 -0400 Subject: [PATCH 27/32] P240: fix EOF settings --- properties/P000240.md | 68 +++++++++++++++++++++---------------------- 1 file changed, 34 insertions(+), 34 deletions(-) diff --git a/properties/P000240.md b/properties/P000240.md index d3a510b8d0..68321c543d 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -1,34 +1,34 @@ ---- -uid: P000240 -name: CW complex -refs: - - wikipedia: CW_complex - name: CW complex on Wikipedia - - zb: "1044.55001" - name: Algebraic Topology (Hatcher) ---- - -$X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots$, such that: - -- $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. -- $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. - -Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) /{\sim}$, -where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^{n+1}$. -The set $J$ is allowed to be empty, in which case $X_{n+1}=X_n$. -The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is empty. - -Here $D^n$ is the closed unit disk in $\mathbb R^n$ with its "boundary" $\partial D^n=S^{n-1}$ being the unit sphere in $\mathbb R^n$. In particular, $D^0=\{0\}$ and $\partial D^0=S^{-1}=\emptyset$ by convention. - -*Note*: A *CW-structure* on a topological space $X$ is a filtration $X_{-1}\subseteq X_0\subseteq X_1 \subseteq\dots$ satisfying the conditions above. -Strictly speaking, a *CW complex* is a space $X$ together with a compatible CW-structure. -For simplicity and by a slight abuse of language, we call the space itself a CW complex if it admits a CW structure. - -An equivalent definition can be given in terms of a decomposition of $X$ into cells and conditions on their associated characteristic maps. -For details and the meaning of this terminology, see Proposition A.2 on page 521 of {{zb:1044.55001}} or Definition II.1.1 in {{zb:0207.21704}}. -For an alternative, categorical definition, see . - ----- -#### Meta-properties - -- This property is preserved by arbitrary disjoint unions. +--- +uid: P000240 +name: CW complex +refs: + - wikipedia: CW_complex + name: CW complex on Wikipedia + - zb: "1044.55001" + name: Algebraic Topology (Hatcher) +--- + +$X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots$, such that: + +- $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. +- $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. + +Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) /{\sim}$, +where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^{n+1}$. +The set $J$ is allowed to be empty, in which case $X_{n+1}=X_n$. +The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is empty. + +Here $D^n$ is the closed unit disk in $\mathbb R^n$ with its "boundary" $\partial D^n=S^{n-1}$ being the unit sphere in $\mathbb R^n$. In particular, $D^0=\{0\}$ and $\partial D^0=S^{-1}=\emptyset$ by convention. + +*Note*: A *CW-structure* on a topological space $X$ is a filtration $X_{-1}\subseteq X_0\subseteq X_1 \subseteq\dots$ satisfying the conditions above. +Strictly speaking, a *CW complex* is a space $X$ together with a compatible CW-structure. +For simplicity and by a slight abuse of language, we call the space itself a CW complex if it admits a CW structure. + +An equivalent definition can be given in terms of a decomposition of $X$ into cells and conditions on their associated characteristic maps. +For details and the meaning of this terminology, see Proposition A.2 on page 521 of {{zb:1044.55001}} or Definition II.1.1 in {{zb:0207.21704}}. +For an alternative, categorical definition, see . + +---- +#### Meta-properties + +- This property is preserved by arbitrary disjoint unions. From 8d6a86f85485691bb53a77f1108341bbe64dbb6c Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Wed, 29 Apr 2026 16:23:20 -0400 Subject: [PATCH 28/32] P240: more references --- properties/P000240.md | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index 68321c543d..5a998835ce 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -6,6 +6,8 @@ refs: name: CW complex on Wikipedia - zb: "1044.55001" name: Algebraic Topology (Hatcher) + - zb: "0207.21704" + name: The Topology of CW Complexes (Lundell & Weingram) --- $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots$, such that: @@ -24,9 +26,12 @@ Here $D^n$ is the closed unit disk in $\mathbb R^n$ with its "boundary" $\partia Strictly speaking, a *CW complex* is a space $X$ together with a compatible CW-structure. For simplicity and by a slight abuse of language, we call the space itself a CW complex if it admits a CW structure. +Defined on page 5 of {{zb:1044.55001}}, also given in Theorem II.2.4 of {{zb:0207.21704}}. +See also . + +---- An equivalent definition can be given in terms of a decomposition of $X$ into cells and conditions on their associated characteristic maps. For details and the meaning of this terminology, see Proposition A.2 on page 521 of {{zb:1044.55001}} or Definition II.1.1 in {{zb:0207.21704}}. -For an alternative, categorical definition, see . ---- #### Meta-properties From 645e351aaf5233a7918bf80e2679fc3ed5862506 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Thu, 30 Apr 2026 02:29:47 +0200 Subject: [PATCH 29/32] move wiki to bottom --- properties/P000240.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/properties/P000240.md b/properties/P000240.md index 5a998835ce..5b4b4a2aea 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -2,12 +2,12 @@ uid: P000240 name: CW complex refs: - - wikipedia: CW_complex - name: CW complex on Wikipedia - zb: "1044.55001" name: Algebraic Topology (Hatcher) - zb: "0207.21704" name: The Topology of CW Complexes (Lundell & Weingram) + - wikipedia: CW_complex + name: CW complex on Wikipedia --- $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots$, such that: From b0aedb74cec529e0cd39f27a80d506bc2634b09a Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 1 May 2026 02:44:04 +0200 Subject: [PATCH 30/32] Update properties/P000240.md Co-authored-by: Geoffrey Sangston --- properties/P000240.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index 5b4b4a2aea..1af1436a10 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -20,7 +20,7 @@ where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for al The set $J$ is allowed to be empty, in which case $X_{n+1}=X_n$. The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is empty. -Here $D^n$ is the closed unit disk in $\mathbb R^n$ with its "boundary" $\partial D^n=S^{n-1}$ being the unit sphere in $\mathbb R^n$. In particular, $D^0=\{0\}$ and $\partial D^0=S^{-1}=\emptyset$ by convention. +Here $D^n$ is the closed unit disk in $\mathbb R^n$ and $\partial D^n=S^{n-1}$ is the unit sphere in $\mathbb R^n$. We set $\partial D^0=S^{-1}=\emptyset$ by convention. *Note*: A *CW-structure* on a topological space $X$ is a filtration $X_{-1}\subseteq X_0\subseteq X_1 \subseteq\dots$ satisfying the conditions above. Strictly speaking, a *CW complex* is a space $X$ together with a compatible CW-structure. From ccc0000eecb60c9f309a73f8278cb5121d981cdf Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 30 Apr 2026 20:54:18 -0400 Subject: [PATCH 31/32] Remove scare quotes. Change indexing to $n$-cells instead of $(n+1)$-cells --- properties/P000240.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/properties/P000240.md b/properties/P000240.md index 1af1436a10..d79ec41367 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -12,12 +12,12 @@ refs: $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots$, such that: -- $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells. -- $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. +- $X_n$ is obtained from $X_{n-1}$ by attaching $n$-cells. +- $X = \bigcup_{n\geq 0} X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. -Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) /{\sim}$, -where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^{n+1}$. -The set $J$ is allowed to be empty, in which case $X_{n+1}=X_n$. +Here *attaching n-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^n \to X_{n-1}$, such that $X_n$ is homeomorphic to the quotient $(X_{n-1} \sqcup (J \times D^n)) /{\sim}$, +where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^n$. +The set $J$ is allowed to be empty, in which case $X_n=X_{n-1}$. The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is empty. Here $D^n$ is the closed unit disk in $\mathbb R^n$ and $\partial D^n=S^{n-1}$ is the unit sphere in $\mathbb R^n$. We set $\partial D^0=S^{-1}=\emptyset$ by convention. From 16b055db2f3d465cdbd92215b4c68a7574383b75 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 1 May 2026 03:29:32 +0200 Subject: [PATCH 32/32] Apply suggestion from @felixpernegger --- properties/P000240.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index d79ec41367..4dd91a8b0d 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -15,7 +15,7 @@ $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subsete - $X_n$ is obtained from $X_{n-1}$ by attaching $n$-cells. - $X = \bigcup_{n\geq 0} X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$. -Here *attaching n-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^n \to X_{n-1}$, such that $X_n$ is homeomorphic to the quotient $(X_{n-1} \sqcup (J \times D^n)) /{\sim}$, +Here *attaching $n$-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^n \to X_{n-1}$, such that $X_n$ is homeomorphic to the quotient $(X_{n-1} \sqcup (J \times D^n)) /{\sim}$, where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^n$. The set $J$ is allowed to be empty, in which case $X_n=X_{n-1}$. The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is empty.