From 873290a518d1b1a75b9f890c331604b631d6730b Mon Sep 17 00:00:00 2001 From: Batixx Date: Wed, 18 Mar 2026 15:44:25 +0100 Subject: [PATCH 1/5] empty --- spaces/S000174/properties/P000023.md | 7 +++++++ spaces/S000174/properties/P000026.md | 7 +++++++ spaces/S000174/properties/P000048.md | 7 +++++++ spaces/S000174/properties/P000051.md | 7 +++++++ spaces/S000174/properties/P000058.md | 7 +++++++ spaces/S000174/properties/P000093.md | 7 +++++++ spaces/S000174/properties/P000139.md | 7 +++++++ spaces/S000174/properties/P000197.md | 7 +++++++ 8 files changed, 56 insertions(+) create mode 100644 spaces/S000174/properties/P000023.md create mode 100644 spaces/S000174/properties/P000026.md create mode 100644 spaces/S000174/properties/P000048.md create mode 100644 spaces/S000174/properties/P000051.md create mode 100644 spaces/S000174/properties/P000058.md create mode 100644 spaces/S000174/properties/P000093.md create mode 100644 spaces/S000174/properties/P000139.md create mode 100644 spaces/S000174/properties/P000197.md diff --git a/spaces/S000174/properties/P000023.md b/spaces/S000174/properties/P000023.md new file mode 100644 index 0000000000..54dacda64d --- /dev/null +++ b/spaces/S000174/properties/P000023.md @@ -0,0 +1,7 @@ +--- +space: S000174 +property: P000023 +value: false +--- + +. diff --git a/spaces/S000174/properties/P000026.md b/spaces/S000174/properties/P000026.md new file mode 100644 index 0000000000..ed58990719 --- /dev/null +++ b/spaces/S000174/properties/P000026.md @@ -0,0 +1,7 @@ +--- +space: S000174 +property: P000026 +value: false +--- + +. diff --git a/spaces/S000174/properties/P000048.md b/spaces/S000174/properties/P000048.md new file mode 100644 index 0000000000..78f089ad1f --- /dev/null +++ b/spaces/S000174/properties/P000048.md @@ -0,0 +1,7 @@ +--- +space: S000174 +property: P000048 +value: true +--- + +. diff --git a/spaces/S000174/properties/P000051.md b/spaces/S000174/properties/P000051.md new file mode 100644 index 0000000000..fdef195ab8 --- /dev/null +++ b/spaces/S000174/properties/P000051.md @@ -0,0 +1,7 @@ +--- +space: S000174 +property: P000051 +value: false +--- + +. diff --git a/spaces/S000174/properties/P000058.md b/spaces/S000174/properties/P000058.md new file mode 100644 index 0000000000..98e129166e --- /dev/null +++ b/spaces/S000174/properties/P000058.md @@ -0,0 +1,7 @@ +--- +space: S000174 +property: P000058 +value: false +--- + +. diff --git a/spaces/S000174/properties/P000093.md b/spaces/S000174/properties/P000093.md new file mode 100644 index 0000000000..5f188fcc4d --- /dev/null +++ b/spaces/S000174/properties/P000093.md @@ -0,0 +1,7 @@ +--- +space: S000174 +property: P000093 +value: false +--- + +. diff --git a/spaces/S000174/properties/P000139.md b/spaces/S000174/properties/P000139.md new file mode 100644 index 0000000000..11cdb2a2e3 --- /dev/null +++ b/spaces/S000174/properties/P000139.md @@ -0,0 +1,7 @@ +--- +space: S000174 +property: P000139 +value: true +--- + +. diff --git a/spaces/S000174/properties/P000197.md b/spaces/S000174/properties/P000197.md new file mode 100644 index 0000000000..aa6e657b16 --- /dev/null +++ b/spaces/S000174/properties/P000197.md @@ -0,0 +1,7 @@ +--- +space: S000174 +property: P000197 +value: false +--- + +. From 43431044c8ca3cc75ed9e7bff5b1edcdd09a8a44 Mon Sep 17 00:00:00 2001 From: Batixx Date: Wed, 18 Mar 2026 16:53:44 +0100 Subject: [PATCH 2/5] description --- spaces/S000174/properties/P000023.md | 2 +- spaces/S000174/properties/P000026.md | 2 +- spaces/S000174/properties/P000048.md | 2 +- spaces/S000174/properties/P000051.md | 2 +- spaces/S000174/properties/P000058.md | 2 +- spaces/S000174/properties/P000093.md | 2 +- spaces/S000174/properties/P000139.md | 2 +- spaces/S000174/properties/P000197.md | 2 +- 8 files changed, 8 insertions(+), 8 deletions(-) diff --git a/spaces/S000174/properties/P000023.md b/spaces/S000174/properties/P000023.md index 54dacda64d..0ac86d0a1f 100644 --- a/spaces/S000174/properties/P000023.md +++ b/spaces/S000174/properties/P000023.md @@ -4,4 +4,4 @@ property: P000023 value: false --- -. +The point $(\omega, \omega, \dots) \in X$ has no compact neighborhood. diff --git a/spaces/S000174/properties/P000026.md b/spaces/S000174/properties/P000026.md index ed58990719..c763432601 100644 --- a/spaces/S000174/properties/P000026.md +++ b/spaces/S000174/properties/P000026.md @@ -4,4 +4,4 @@ property: P000026 value: false --- -. +Let $N$ be a countable subset of $X$. Let $r := \sup \{x_0 \mid x \in N\}$. Then the point $(r + 1,0,0,\dots)$ has the neighborhood $\{r + 1\} \times \omega_1^\omega$ which is disjoint with $N$, so $N$ is not dense. diff --git a/spaces/S000174/properties/P000048.md b/spaces/S000174/properties/P000048.md index 78f089ad1f..0db385021b 100644 --- a/spaces/S000174/properties/P000048.md +++ b/spaces/S000174/properties/P000048.md @@ -4,4 +4,4 @@ property: P000048 value: true --- -. +Let diff --git a/spaces/S000174/properties/P000051.md b/spaces/S000174/properties/P000051.md index fdef195ab8..0d59737677 100644 --- a/spaces/S000174/properties/P000051.md +++ b/spaces/S000174/properties/P000051.md @@ -4,4 +4,4 @@ property: P000051 value: false --- -. +The subspace $2 ^ \omega \setminus \{(0,0,\dots)\} \subseteq X$ has no isolated point. diff --git a/spaces/S000174/properties/P000058.md b/spaces/S000174/properties/P000058.md index 98e129166e..6de00163d6 100644 --- a/spaces/S000174/properties/P000058.md +++ b/spaces/S000174/properties/P000058.md @@ -4,4 +4,4 @@ property: P000058 value: false --- -. +$|X|=|\omega_1^\omega| \geq |2^\omega| = \mathfrak{c}$ diff --git a/spaces/S000174/properties/P000093.md b/spaces/S000174/properties/P000093.md index 5f188fcc4d..7b68071754 100644 --- a/spaces/S000174/properties/P000093.md +++ b/spaces/S000174/properties/P000093.md @@ -4,4 +4,4 @@ property: P000093 value: false --- -. +The point $(\omega, \omega,\dots) \in X$ has no countable neighborhood. diff --git a/spaces/S000174/properties/P000139.md b/spaces/S000174/properties/P000139.md index 11cdb2a2e3..2c3f076b77 100644 --- a/spaces/S000174/properties/P000139.md +++ b/spaces/S000174/properties/P000139.md @@ -4,4 +4,4 @@ property: P000139 value: true --- -. +The point $p := (0, 0, \dots) \in X$ is isolated via $\{p\} = 0 ^ \omega \cap \omega_1 ^ \omega$. diff --git a/spaces/S000174/properties/P000197.md b/spaces/S000174/properties/P000197.md index aa6e657b16..d5623fcbf9 100644 --- a/spaces/S000174/properties/P000197.md +++ b/spaces/S000174/properties/P000197.md @@ -4,4 +4,4 @@ property: P000197 value: false --- -. +The subspace $\omega_1 \times 0^\omega \subseteq X$ is uncountable and discrete. From b0c34ceb1f15326aaba3052423cd545da2f50293 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Thu, 26 Mar 2026 17:34:44 +0100 Subject: [PATCH 3/5] delete p48 for now --- spaces/S000174/properties/P000048.md | 7 ------- 1 file changed, 7 deletions(-) delete mode 100644 spaces/S000174/properties/P000048.md diff --git a/spaces/S000174/properties/P000048.md b/spaces/S000174/properties/P000048.md deleted file mode 100644 index 0db385021b..0000000000 --- a/spaces/S000174/properties/P000048.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000174 -property: P000048 -value: true ---- - -Let From 369a72778443802cb660b93686a2729d9fd4b9bd Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Thu, 13 Aug 2026 01:21:58 +0200 Subject: [PATCH 4/5] Apply suggestions from code review Co-authored-by: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> --- spaces/S000174/properties/P000051.md | 2 +- spaces/S000174/properties/P000139.md | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/spaces/S000174/properties/P000051.md b/spaces/S000174/properties/P000051.md index 0d59737677..25f7a2e0c0 100644 --- a/spaces/S000174/properties/P000051.md +++ b/spaces/S000174/properties/P000051.md @@ -4,4 +4,4 @@ property: P000051 value: false --- -The subspace $2 ^ \omega \setminus \{(0,0,\dots)\} \subseteq X$ has no isolated point. +The subspace $2 ^ \omega \setminus \{(0,0,\dots)\} \subseteq X$ is homeomorphic to Cantor set without a point, and so has no isolated points. diff --git a/spaces/S000174/properties/P000139.md b/spaces/S000174/properties/P000139.md index 2c3f076b77..69d03db20e 100644 --- a/spaces/S000174/properties/P000139.md +++ b/spaces/S000174/properties/P000139.md @@ -4,4 +4,4 @@ property: P000139 value: true --- -The point $p := (0, 0, \dots) \in X$ is isolated via $\{p\} = 0 ^ \omega \cap \omega_1 ^ \omega$. +The point $p := (0, 0, \dots) \in X$ is isolated since $\{p\} = 1^\omega$ where $1 = \{0\}$. From 7322d38371a349337585d16db9443ca14c98dad9 Mon Sep 17 00:00:00 2001 From: Batixx Date: Thu, 13 Aug 2026 02:46:21 +0200 Subject: [PATCH 5/5] apply suggestions --- spaces/S000174/properties/P000023.md | 6 +++++- spaces/S000174/properties/P000026.md | 7 ------- spaces/S000174/properties/P000058.md | 7 ------- spaces/S000174/properties/P000062.md | 10 ++++++++++ spaces/S000174/properties/P000065.md | 9 +++++++++ spaces/S000174/properties/P000083.md | 16 ++++++++++++++++ spaces/S000174/properties/P000163.md | 7 ------- spaces/S000174/properties/P000194.md | 16 ++++++++++++++++ spaces/S000174/properties/P000197.md | 7 ------- 9 files changed, 56 insertions(+), 29 deletions(-) delete mode 100644 spaces/S000174/properties/P000026.md delete mode 100644 spaces/S000174/properties/P000058.md create mode 100644 spaces/S000174/properties/P000062.md create mode 100644 spaces/S000174/properties/P000065.md create mode 100644 spaces/S000174/properties/P000083.md delete mode 100644 spaces/S000174/properties/P000163.md create mode 100644 spaces/S000174/properties/P000194.md delete mode 100644 spaces/S000174/properties/P000197.md diff --git a/spaces/S000174/properties/P000023.md b/spaces/S000174/properties/P000023.md index 0ac86d0a1f..1154b6372e 100644 --- a/spaces/S000174/properties/P000023.md +++ b/spaces/S000174/properties/P000023.md @@ -4,4 +4,8 @@ property: P000023 value: false --- -The point $(\omega, \omega, \dots) \in X$ has no compact neighborhood. +At the point $p=(\omega, \omega, \dots)$, the clopen sets +$\{\omega\}^n \times (\omega+1)^\omega$ form a neighborhood base. Each of +these sets has a topology finer than the product topology on +$\{\omega\}^n \times (\omega+1)^\omega$ when $\omega+1$ is discrete. The +latter space is not compact, so none of these neighborhoods is compact. diff --git a/spaces/S000174/properties/P000026.md b/spaces/S000174/properties/P000026.md deleted file mode 100644 index c763432601..0000000000 --- a/spaces/S000174/properties/P000026.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000174 -property: P000026 -value: false ---- - -Let $N$ be a countable subset of $X$. Let $r := \sup \{x_0 \mid x \in N\}$. Then the point $(r + 1,0,0,\dots)$ has the neighborhood $\{r + 1\} \times \omega_1^\omega$ which is disjoint with $N$, so $N$ is not dense. diff --git a/spaces/S000174/properties/P000058.md b/spaces/S000174/properties/P000058.md deleted file mode 100644 index 6de00163d6..0000000000 --- a/spaces/S000174/properties/P000058.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000174 -property: P000058 -value: false ---- - -$|X|=|\omega_1^\omega| \geq |2^\omega| = \mathfrak{c}$ diff --git a/spaces/S000174/properties/P000062.md b/spaces/S000174/properties/P000062.md new file mode 100644 index 0000000000..672208b51a --- /dev/null +++ b/spaces/S000174/properties/P000062.md @@ -0,0 +1,10 @@ +--- +space: S000174 +property: P000062 +value: false +--- + +The collection $\{\xi^\omega : \xi < \omega_1\}$ is an open cover of $X$. +Every countable subcollection has union contained in $\xi^\omega$ for some +$\xi < \omega_1$, and $\xi^\omega$ is a proper closed subset of $X$. Thus, +no countable subcollection has dense union. diff --git a/spaces/S000174/properties/P000065.md b/spaces/S000174/properties/P000065.md new file mode 100644 index 0000000000..69fecf56c4 --- /dev/null +++ b/spaces/S000174/properties/P000065.md @@ -0,0 +1,9 @@ +--- +space: S000174 +property: P000065 +value: true +--- + + +\[\mathfrak c = |2^\omega| \leq |\omega_1^\omega| +\leq |(2^\omega)^\omega| = \mathfrak c\] diff --git a/spaces/S000174/properties/P000083.md b/spaces/S000174/properties/P000083.md new file mode 100644 index 0000000000..6be284efd6 --- /dev/null +++ b/spaces/S000174/properties/P000083.md @@ -0,0 +1,16 @@ +--- +space: S000174 +property: P000083 +value: false +refs: + - doi: 10.4064/FM-97-2-53-55 + name: "A perfectly normal locally metrizable non-paracompact space" + - zb: "0593.28016" + name: Borel measures (Gardner and Pfeffer) +--- + +Proposition 2 of {{doi:10.4064/FM-97-2-53-55}} constructs a locally countable +subspace of $X$ which is not $\sigma$-discrete. By Theorem 13.3 of +{{zb:0593.28016}}, every locally countable subspace of +a weakly $\delta\theta$-refinable space is $\sigma$-discrete. Hence $X$ is not +weakly $\delta\theta$-refinable and, in particular, is not meta-Lindelöf. diff --git a/spaces/S000174/properties/P000163.md b/spaces/S000174/properties/P000163.md deleted file mode 100644 index 03beb65bf6..0000000000 --- a/spaces/S000174/properties/P000163.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000174 -property: P000163 -value: true ---- - -$|X| = |\omega_1^\omega| \leq (2^\omega)^\omega = 2^\omega = \mathfrak{c}$. diff --git a/spaces/S000174/properties/P000194.md b/spaces/S000174/properties/P000194.md new file mode 100644 index 0000000000..7a3456cb91 --- /dev/null +++ b/spaces/S000174/properties/P000194.md @@ -0,0 +1,16 @@ +--- +space: S000174 +property: P000194 +value: false +refs: + - doi: 10.4064/FM-97-2-53-55 + name: "A perfectly normal locally metrizable non-paracompact space" + - zb: "0593.28016" + name: Borel measures (Gardner and Pfeffer) +--- + +Proposition 2 of {{doi:10.4064/FM-97-2-53-55}} constructs a locally countable +subspace of $X$ which is not $\sigma$-discrete. By Theorem 13.3 of +{{zb:0593.28016}}, every locally countable subspace of +a weakly $\delta\theta$-refinable space is $\sigma$-discrete. Hence $X$ is not +weakly $\delta\theta$-refinable and, in particular, is not submetacompact. diff --git a/spaces/S000174/properties/P000197.md b/spaces/S000174/properties/P000197.md deleted file mode 100644 index d5623fcbf9..0000000000 --- a/spaces/S000174/properties/P000197.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000174 -property: P000197 -value: false ---- - -The subspace $\omega_1 \times 0^\omega \subseteq X$ is uncountable and discrete.