From 636527970bb1523a8fa76b63d284c9c0fc427a35 Mon Sep 17 00:00:00 2001 From: Patrick Rabau Date: Sat, 28 Sep 2024 12:37:03 -0400 Subject: [PATCH] More aleph_0 spaces --- spaces/S000131/properties/P000023.md | 12 ------------ spaces/S000131/properties/P000183.md | 14 ++++++++++++++ spaces/S000139/properties/P000023.md | 10 ---------- spaces/S000139/properties/P000026.md | 7 ------- spaces/S000139/properties/P000064.md | 2 +- spaces/S000139/properties/P000183.md | 10 ++++++++++ spaces/S000156/README.md | 2 +- spaces/S000156/properties/P000023.md | 10 ---------- spaces/S000156/properties/P000080.md | 2 +- spaces/S000156/properties/P000183.md | 10 ++++++++++ 10 files changed, 37 insertions(+), 42 deletions(-) delete mode 100644 spaces/S000131/properties/P000023.md create mode 100644 spaces/S000131/properties/P000183.md delete mode 100644 spaces/S000139/properties/P000023.md delete mode 100644 spaces/S000139/properties/P000026.md create mode 100644 spaces/S000139/properties/P000183.md delete mode 100644 spaces/S000156/properties/P000023.md create mode 100644 spaces/S000156/properties/P000183.md diff --git a/spaces/S000131/properties/P000023.md b/spaces/S000131/properties/P000023.md deleted file mode 100644 index 5442de7cc..000000000 --- a/spaces/S000131/properties/P000023.md +++ /dev/null @@ -1,12 +0,0 @@ ---- -space: S000131 -property: P000023 -value: false -refs: - - mathse: 4844426 - name: Answer to "Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?" ---- - -The sets $K_n=([0,n]\times\omega)\cup\{\infty\}$ are compact and every compact set in $X$ is contained in one of the $K_n$. But no $K_n$ is a neighborhood of $\infty$, so $\infty$ has no compact neighborhood. - -See {{mathse:4844426}}. diff --git a/spaces/S000131/properties/P000183.md b/spaces/S000131/properties/P000183.md new file mode 100644 index 000000000..7e59be3ad --- /dev/null +++ b/spaces/S000131/properties/P000183.md @@ -0,0 +1,14 @@ +--- +space: S000131 +property: P000183 +value: true +refs: + - mathse: 4844426 + name: Answer to "Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?" +--- + +Each spine $C_m=(\{m\}\times\omega)\cup\{\infty\}$ is a closed subspace of $X$ homeomorphic to a convergent sequence ({S20}); +and {S20|P183}. +And every compact subset of $X$ is contained in the union of a finite number of these spines. + +Therefore, taking a countable $k$-network from each of the (countably many) spines and forming their union gives a countable $k$-network for $X.$ diff --git a/spaces/S000139/properties/P000023.md b/spaces/S000139/properties/P000023.md deleted file mode 100644 index 06fc4093e..000000000 --- a/spaces/S000139/properties/P000023.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000139 -property: P000023 -value: false -refs: - - mathse: 4844916 - name: Answer to "Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?" ---- - -See {{mathse:4844916}}. diff --git a/spaces/S000139/properties/P000026.md b/spaces/S000139/properties/P000026.md deleted file mode 100644 index 9fcfd7a5e..000000000 --- a/spaces/S000139/properties/P000026.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000139 -property: P000026 -value: true ---- - -$X$ is a continuous image of $\mathbb R$, which is {P26}. diff --git a/spaces/S000139/properties/P000064.md b/spaces/S000139/properties/P000064.md index 3ebf245fd..801f3b3ad 100644 --- a/spaces/S000139/properties/P000064.md +++ b/spaces/S000139/properties/P000064.md @@ -7,4 +7,4 @@ refs: name: Baire space on Wikipedia --- -The subspace $X\setminus\{\infty\}$ is Baire and dense in $X$. +The subspace $X\setminus\{\infty\}$ is Baire (because locally compact Hausdorff) and dense in $X$. diff --git a/spaces/S000139/properties/P000183.md b/spaces/S000139/properties/P000183.md new file mode 100644 index 000000000..b97344c4a --- /dev/null +++ b/spaces/S000139/properties/P000183.md @@ -0,0 +1,10 @@ +--- +space: S000139 +property: P000183 +value: true +--- + +Each of the circles (corresponding to an interval $[n,n+1]$, $n\in\mathbb Z$, with the endpoints identified) is a closed subspace of $X$ and {S170|P183}. +And every compact subset of $X$ is contained in the union of a finite number of these circles. + +Therefore, taking a countable $k$-network from each of the (countably many) circles and forming their union gives a countable $k$-network for $X.$ diff --git a/spaces/S000156/README.md b/spaces/S000156/README.md index 82e41b3ce..92d1f20fd 100644 --- a/spaces/S000156/README.md +++ b/spaces/S000156/README.md @@ -2,7 +2,7 @@ uid: S000156 name: Arens space aliases: -- S_2 +- $S_2$ refs: - mr: 37500 name: Note on convergence in topology (Arens) diff --git a/spaces/S000156/properties/P000023.md b/spaces/S000156/properties/P000023.md deleted file mode 100644 index 360bf6db0..000000000 --- a/spaces/S000156/properties/P000023.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000156 -property: P000023 -value: false -refs: - - mathse: 4672783 - name: Is the Arens space hemicompact but not locally compact? ---- - -See {{mathse:4672783}}. diff --git a/spaces/S000156/properties/P000080.md b/spaces/S000156/properties/P000080.md index 02a55ca1a..b9b5e38dc 100644 --- a/spaces/S000156/properties/P000080.md +++ b/spaces/S000156/properties/P000080.md @@ -9,4 +9,4 @@ refs: The Arens space contains {S23} as a subspace which is not {P80}. Since {P80} is a hereditary property, the Arens space does not satisfy the property either. -See https://dantopology.wordpress.com/2010/08/18/a-note-about-the-arens-space/ +See . diff --git a/spaces/S000156/properties/P000183.md b/spaces/S000156/properties/P000183.md new file mode 100644 index 000000000..cc8826c16 --- /dev/null +++ b/spaces/S000156/properties/P000183.md @@ -0,0 +1,10 @@ +--- +space: S000156 +property: P000183 +value: true +refs: + - mathse: 4673444 + name: Answer to "Is the Arens space hemicompact but not locally compact?" +--- + +See the last part of {{mathse:4673444}}.