From ca06df9f232d02bb94791affaf50be1c8f57d228 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 24 May 2026 09:20:29 +0200 Subject: [PATCH 1/4] whitehead theorem --- spaces/S000156/README.md | 16 ++++++++++++++-- theorems/T000893.md | 16 ++++++++++++++++ 2 files changed, 30 insertions(+), 2 deletions(-) create mode 100644 theorems/T000893.md diff --git a/spaces/S000156/README.md b/spaces/S000156/README.md index 92d1f20fdd..c618e43800 100644 --- a/spaces/S000156/README.md +++ b/spaces/S000156/README.md @@ -15,6 +15,18 @@ countable discrete space. Arens space $S_2$ is the minimal topology on $(X\times\omega)\cup\{\infty'\}$ where $(\{\infty\}\times\omega)\cup\{\infty'\}$ is homeomorphic to $X$. -This space was originally defined by Richard Arens in {{mr:0037500}}. +Note the subspace $(\omega\times\omega)\cup\{\infty'\}$ of the Arens space is homeomorphic to {S23}, as mentioned in Examples 2.10 and 3.10 of {{mr:3269014}}. + +Alternatively, let $\mathbb{N}$ be the set of all positive integers. In this formulation, the Arens' space is the set +$X = \{\infty\} \cup \mathbb{N} \cup (\mathbb{N} \times \mathbb{N})$ with the open neighborhoods defined according to the following conditions: -The subspace $(\omega\times\omega)\cup\{\infty'\}$ of the Arens space is homeomorphic to {S23}, as mentioned in Examples 2.10 and 3.10 of {{mr:3269014}}. +- The points in $\mathbb{N} \times \mathbb{N}$ are isolated +- The neighborhoods at each $n \in \mathbb{N}$ are of the form + \[ + B_{n,m} = \{n\} \cup \{(n,j) \in \mathbb{N} \times \mathbb{N} : j \geq m\} + \] + for some $m \in \mathbb{N}$; +- The neighborhoods at $\infty$ are obtained by removing from $X$ finitely many $B_{n,1}$ + and by removing finitely many isolated points in each of the remaining $B_{n,1}$. + +This space was originally defined by Richard Arens in {{mr:0037500}}. diff --git a/theorems/T000893.md b/theorems/T000893.md new file mode 100644 index 0000000000..7d386f0105 --- /dev/null +++ b/theorems/T000893.md @@ -0,0 +1,16 @@ +--- +uid: T000893 +if: + and: + - P000240: true + - P000242: true +then: + P000199: false +refs: + - wikipedia: Whitehead_theorem + name: Whitehead theorem on Wikipedia + - zb: "1044.55001" + name: Algebraic Topology (Hatcher) +--- + +Since {S162|P240}, $X$ is homotopy equivalent to the singleton by the [Whitehead theorem](https://en.wikipedia.org/wiki/Whitehead_theorem). From 9df1804898f926fd3a9f494eafe8b96444e59834 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 24 May 2026 09:21:39 +0200 Subject: [PATCH 2/4] remove thing from other branch --- spaces/S000156/README.md | 16 ++-------------- 1 file changed, 2 insertions(+), 14 deletions(-) diff --git a/spaces/S000156/README.md b/spaces/S000156/README.md index c618e43800..92d1f20fdd 100644 --- a/spaces/S000156/README.md +++ b/spaces/S000156/README.md @@ -15,18 +15,6 @@ countable discrete space. Arens space $S_2$ is the minimal topology on $(X\times\omega)\cup\{\infty'\}$ where $(\{\infty\}\times\omega)\cup\{\infty'\}$ is homeomorphic to $X$. -Note the subspace $(\omega\times\omega)\cup\{\infty'\}$ of the Arens space is homeomorphic to {S23}, as mentioned in Examples 2.10 and 3.10 of {{mr:3269014}}. - -Alternatively, let $\mathbb{N}$ be the set of all positive integers. In this formulation, the Arens' space is the set -$X = \{\infty\} \cup \mathbb{N} \cup (\mathbb{N} \times \mathbb{N})$ with the open neighborhoods defined according to the following conditions: - -- The points in $\mathbb{N} \times \mathbb{N}$ are isolated -- The neighborhoods at each $n \in \mathbb{N}$ are of the form - \[ - B_{n,m} = \{n\} \cup \{(n,j) \in \mathbb{N} \times \mathbb{N} : j \geq m\} - \] - for some $m \in \mathbb{N}$; -- The neighborhoods at $\infty$ are obtained by removing from $X$ finitely many $B_{n,1}$ - and by removing finitely many isolated points in each of the remaining $B_{n,1}$. - This space was originally defined by Richard Arens in {{mr:0037500}}. + +The subspace $(\omega\times\omega)\cup\{\infty'\}$ of the Arens space is homeomorphic to {S23}, as mentioned in Examples 2.10 and 3.10 of {{mr:3269014}}. From 399d7a9a76d61058eec90e2390ffc85c41d6f09f Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Mon, 25 May 2026 09:58:40 +0200 Subject: [PATCH 3/4] fix statement --- theorems/T000893.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000893.md b/theorems/T000893.md index 7d386f0105..0f19e0041e 100644 --- a/theorems/T000893.md +++ b/theorems/T000893.md @@ -5,7 +5,7 @@ if: - P000240: true - P000242: true then: - P000199: false + P000199: true refs: - wikipedia: Whitehead_theorem name: Whitehead theorem on Wikipedia From e617a8074d07fa63126c19982355971d2c81995c Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 26 May 2026 08:58:37 +0200 Subject: [PATCH 4/4] Update theorems/T000893.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000893.md | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/theorems/T000893.md b/theorems/T000893.md index 0f19e0041e..95be921d4b 100644 --- a/theorems/T000893.md +++ b/theorems/T000893.md @@ -13,4 +13,11 @@ refs: name: Algebraic Topology (Hatcher) --- -Since {S162|P240}, $X$ is homotopy equivalent to the singleton by the [Whitehead theorem](https://en.wikipedia.org/wiki/Whitehead_theorem). +The [Whitehead theorem](https://en.wikipedia.org/wiki/Whitehead_theorem) states that a +[weak homotopy equivalence](https://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory)) +from one CW complex to another is a homotopy equivalence. +See Theorem 4.5 in {{zb:1044.55001}}. + +In particular, suppose $X$ is a {P240}. Let $Y$ be {S162} and let $f:X\to Y$ be the constant map to $Y$. +If $X$ is {P242}, the map $f$ is a weak homotopy equivalence. +Since {S162|P240}, the Whitehead theorem implies that $X$ is homotopy equivalent to $Y$; that is, $X$ is {P199}.