From 0a4b1b0e870eba5d4440122c309e9f468e6657e6 Mon Sep 17 00:00:00 2001 From: Batixx Date: Fri, 1 May 2026 05:10:05 +0200 Subject: [PATCH 1/5] discrete cw complex --- properties/P000240.md | 4 ++-- theorems/T000889.md | 9 +++++++++ 2 files changed, 11 insertions(+), 2 deletions(-) create mode 100644 theorems/T000889.md diff --git a/properties/P000240.md b/properties/P000240.md index 4dd91a8b0..ebf3fc692 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -22,8 +22,8 @@ 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 $\mathbb R^n$ 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. +*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. Defined on page 5 of {{zb:1044.55001}}, also given in Theorem II.2.4 of {{zb:0207.21704}}. diff --git a/theorems/T000889.md b/theorems/T000889.md new file mode 100644 index 000000000..638615f1c --- /dev/null +++ b/theorems/T000889.md @@ -0,0 +1,9 @@ +--- +uid: T000889 +if: + P000052: true +then: + P000240: true +--- + +The filtration $X_n := X$ for $n \geq 0$ equips $X$ with a CW structure. From de28a06775500cb64f432400090926bd762664db Mon Sep 17 00:00:00 2001 From: Batixx Date: Fri, 1 May 2026 05:17:09 +0200 Subject: [PATCH 2/5] empty --- properties/P000240.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000240.md b/properties/P000240.md index ebf3fc692..484d7c0d7 100644 --- a/properties/P000240.md +++ b/properties/P000240.md @@ -22,7 +22,7 @@ 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 $\mathbb R^n$ 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. +*Note*: A *CW structure* on a topological space $X$ is a filtration $\emptyset = X_{-1}\subseteq X_0\subseteq X_1 \subseteq\dots$ 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. From 70a7378a5992ca244711816522df30642df3efff Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 1 May 2026 06:28:04 +0200 Subject: [PATCH 3/5] Clarify CW structure description in T000889.md --- theorems/T000889.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000889.md b/theorems/T000889.md index 638615f1c..fe487e802 100644 --- a/theorems/T000889.md +++ b/theorems/T000889.md @@ -6,4 +6,4 @@ then: P000240: true --- -The filtration $X_n := X$ for $n \geq 0$ equips $X$ with a CW structure. +The filtration $X_n := X$ for $n \geq 0$ equips $X$ is a valid CW structure for $X$. From b6567c20d1a744090654de4609d7f493bef1198d Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 1 May 2026 07:38:30 +0200 Subject: [PATCH 4/5] Update theorems/T000889.md Co-authored-by: Geoffrey Sangston --- theorems/T000889.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000889.md b/theorems/T000889.md index fe487e802..01b6f4d61 100644 --- a/theorems/T000889.md +++ b/theorems/T000889.md @@ -6,4 +6,4 @@ then: P000240: true --- -The filtration $X_n := X$ for $n \geq 0$ equips $X$ is a valid CW structure for $X$. +The filtration $X_n := X$ for $n \geq 0$ is a CW structure on $X$. From 67dddad296ded5b546830fcb13366cd71f50d7e4 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Sun, 3 May 2026 03:40:00 +0200 Subject: [PATCH 5/5] Update theorems/T000889.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000889.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000889.md b/theorems/T000889.md index 01b6f4d61..e57b772a2 100644 --- a/theorems/T000889.md +++ b/theorems/T000889.md @@ -6,4 +6,4 @@ then: P000240: true --- -The filtration $X_n := X$ for $n \geq 0$ is a CW structure on $X$. +Choose $X_0=X$.