From 4706b6b22f8bc9d14083296efb067d3df19c306e Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 27 Dec 2025 19:53:57 +0100 Subject: [PATCH 01/10] two traits --- spaces/S000035/properties/P000062.md | 7 +++++++ spaces/S000035/properties/P000082.md | 7 +++++++ 2 files changed, 14 insertions(+) create mode 100644 spaces/S000035/properties/P000062.md create mode 100644 spaces/S000035/properties/P000082.md diff --git a/spaces/S000035/properties/P000062.md b/spaces/S000035/properties/P000062.md new file mode 100644 index 000000000..df04c711e --- /dev/null +++ b/spaces/S000035/properties/P000062.md @@ -0,0 +1,7 @@ +--- +space: S000035 +property: P000062 +value: false +--- + +Consider the open cover $\mathcal{O}=\{[0,a)\mid a \in X\}$. If $X$ were weakly Lindelöf, we would find a countable $\mathcal{U}\subseteq \mathcal{O}$ such that $\bigcup \mathcal{U}$ dense. But then $\bigcup\mathcal{U}=X$, since otherwise we would find an $a \in X$ with $[a,\omega_1)\subseteq X \setminus \bigcup\mathcal{U}$. Therefore $X$ would be a countable union of countable sets and thus countable. diff --git a/spaces/S000035/properties/P000082.md b/spaces/S000035/properties/P000082.md new file mode 100644 index 000000000..619dd2435 --- /dev/null +++ b/spaces/S000035/properties/P000082.md @@ -0,0 +1,7 @@ +--- +space: S000035 +property: P000082 +value: true +--- + +For $a \in X$ consider the open neighborhood $U_a := [0,a+1)\subseteq X$. By the definition of $X$, $U_a$ is countable ordinal space and thus metrizable, see [explore](https://topology.pi-base.org/spaces?q=countable+%2B+ordinal+%2B+%7EMetrizable). From 872540163d709e167545d226fe4001737157429f Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 27 Dec 2025 20:01:25 +0100 Subject: [PATCH 02/10] better explore --- spaces/S000035/properties/P000082.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000035/properties/P000082.md b/spaces/S000035/properties/P000082.md index 619dd2435..a0c01e3e5 100644 --- a/spaces/S000035/properties/P000082.md +++ b/spaces/S000035/properties/P000082.md @@ -4,4 +4,4 @@ property: P000082 value: true --- -For $a \in X$ consider the open neighborhood $U_a := [0,a+1)\subseteq X$. By the definition of $X$, $U_a$ is countable ordinal space and thus metrizable, see [explore](https://topology.pi-base.org/spaces?q=countable+%2B+ordinal+%2B+%7EMetrizable). +For $a \in X$ consider the open neighborhood $U_a := [0,a+1)\subseteq X$. By the definition of $X$, $U_a$ is countable ordinal space and thus metrizable, see [explore](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable). From 44e065e5bba691015d4420a166816f6a9621e78f Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Sun, 28 Dec 2025 01:32:22 +0100 Subject: [PATCH 03/10] Update spaces/S000035/properties/P000082.md Co-authored-by: yhx-12243 --- spaces/S000035/properties/P000082.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000035/properties/P000082.md b/spaces/S000035/properties/P000082.md index a0c01e3e5..56c418f30 100644 --- a/spaces/S000035/properties/P000082.md +++ b/spaces/S000035/properties/P000082.md @@ -4,4 +4,4 @@ property: P000082 value: true --- -For $a \in X$ consider the open neighborhood $U_a := [0,a+1)\subseteq X$. By the definition of $X$, $U_a$ is countable ordinal space and thus metrizable, see [explore](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable). +For $a \in X$ consider the open neighborhood $U_a := [0,a+1)\subseteq X$. By the definition of $X$, $U_a$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable). From 3a391f43dff76502b4fd8efb7210c2a237cb293d Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 28 Dec 2025 01:44:38 +0100 Subject: [PATCH 04/10] include p76 --- spaces/S000035/properties/P000076.md | 10 ++++++++++ 1 file changed, 10 insertions(+) create mode 100644 spaces/S000035/properties/P000076.md diff --git a/spaces/S000035/properties/P000076.md b/spaces/S000035/properties/P000076.md new file mode 100644 index 000000000..7ccebf8e6 --- /dev/null +++ b/spaces/S000035/properties/P000076.md @@ -0,0 +1,10 @@ +--- +space: S000035 +property: P000076 +value: true +refs: +- zb: "1375.54007" + name: Uniform Powers of Compacta and the Proximal Game +--- + +See Theorem 6.2 (c) in {{zb:1375.54007}}. From d9f429679165270877a47aec5d0ae1c413c0981e Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Fri, 2 Jan 2026 10:42:44 +0100 Subject: [PATCH 05/10] new meta-properties, new trait --- properties/P000087.md | 1 + properties/P000186.md | 1 + spaces/S000035/properties/P000186.md | 11 +++++++++++ 3 files changed, 13 insertions(+) create mode 100644 spaces/S000035/properties/P000186.md diff --git a/properties/P000087.md b/properties/P000087.md index 218b0ac06..92302e328 100644 --- a/properties/P000087.md +++ b/properties/P000087.md @@ -17,3 +17,4 @@ Contrary to Munkres or Willard, we do not assume any separation axiom like {P3}, #### Meta-properties - This property is preserved by arbitrary products. +- This property is preserved by $\Sigma$-products. diff --git a/properties/P000186.md b/properties/P000186.md index 86224e785..40a361540 100644 --- a/properties/P000186.md +++ b/properties/P000186.md @@ -13,3 +13,4 @@ Homeomorphic to a subspace of a space that is {P87} and {P187}. - This property is hereditary. - This property is preserved by countable products. +- This property is preserved by $\Sigma$-products. diff --git a/spaces/S000035/properties/P000186.md b/spaces/S000035/properties/P000186.md new file mode 100644 index 000000000..0b839757a --- /dev/null +++ b/spaces/S000035/properties/P000186.md @@ -0,0 +1,11 @@ +--- +space: S000035 +property: P000186 +value: true +refs: +- zb: "0559.54009" + name: Products of normal spaces (T. Przymusiński), Ch. 18 of Handbook of set-theoretic topology +--- + +From proposition 7.2 of {{zb:0559.54009}}, +$\omega_1$ embeds into a $\Sigma$-product of real lines and {S25|P186}. From 20bde64d2597bf5595ccb45c688fed7fd0ccf142 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Mon, 5 Jan 2026 01:51:24 +0100 Subject: [PATCH 06/10] delete proximal --- spaces/S000035/properties/P000076.md | 10 ---------- 1 file changed, 10 deletions(-) delete mode 100644 spaces/S000035/properties/P000076.md diff --git a/spaces/S000035/properties/P000076.md b/spaces/S000035/properties/P000076.md deleted file mode 100644 index 7ccebf8e6..000000000 --- a/spaces/S000035/properties/P000076.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000035 -property: P000076 -value: true -refs: -- zb: "1375.54007" - name: Uniform Powers of Compacta and the Proximal Game ---- - -See Theorem 6.2 (c) in {{zb:1375.54007}}. From 0314cf090879608f3c52ca6058f049813252b0ea Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Mon, 5 Jan 2026 01:57:56 +0100 Subject: [PATCH 07/10] delete embeddable W space --- spaces/S000035/properties/P000186.md | 11 ----------- 1 file changed, 11 deletions(-) delete mode 100644 spaces/S000035/properties/P000186.md diff --git a/spaces/S000035/properties/P000186.md b/spaces/S000035/properties/P000186.md deleted file mode 100644 index 0b839757a..000000000 --- a/spaces/S000035/properties/P000186.md +++ /dev/null @@ -1,11 +0,0 @@ ---- -space: S000035 -property: P000186 -value: true -refs: -- zb: "0559.54009" - name: Products of normal spaces (T. Przymusiński), Ch. 18 of Handbook of set-theoretic topology ---- - -From proposition 7.2 of {{zb:0559.54009}}, -$\omega_1$ embeds into a $\Sigma$-product of real lines and {S25|P186}. From 739e0fee11ca9893425175ae22fea7d85361670d Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 6 Jan 2026 23:13:41 +0100 Subject: [PATCH 08/10] Update spaces/S000035/properties/P000082.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- spaces/S000035/properties/P000082.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000035/properties/P000082.md b/spaces/S000035/properties/P000082.md index 56c418f30..2255c05f1 100644 --- a/spaces/S000035/properties/P000082.md +++ b/spaces/S000035/properties/P000082.md @@ -4,4 +4,4 @@ property: P000082 value: true --- -For $a \in X$ consider the open neighborhood $U_a := [0,a+1)\subseteq X$. By the definition of $X$, $U_a$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable). +For $alpha \in X$ consider the open neighborhood $U_\alpha := [0,\alpha+1)\subseteq X$. By the definition of $X$, $U_\alpha$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable). From 2ae925a87c830db71cfa38633455b667c50a7c71 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 6 Jan 2026 23:27:16 +0100 Subject: [PATCH 09/10] Update spaces/S000035/properties/P000062.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- spaces/S000035/properties/P000062.md | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/spaces/S000035/properties/P000062.md b/spaces/S000035/properties/P000062.md index df04c711e..6981cd361 100644 --- a/spaces/S000035/properties/P000062.md +++ b/spaces/S000035/properties/P000062.md @@ -4,4 +4,7 @@ property: P000062 value: false --- -Consider the open cover $\mathcal{O}=\{[0,a)\mid a \in X\}$. If $X$ were weakly Lindelöf, we would find a countable $\mathcal{U}\subseteq \mathcal{O}$ such that $\bigcup \mathcal{U}$ dense. But then $\bigcup\mathcal{U}=X$, since otherwise we would find an $a \in X$ with $[a,\omega_1)\subseteq X \setminus \bigcup\mathcal{U}$. Therefore $X$ would be a countable union of countable sets and thus countable. +Consider the open cover $\mathcal{O}=\{[0,\alpha)\mid \alpha \in X\}$. +Every countable subset of $\omega_1$ has an upper bound in $\omega_1$. +So if $\mathcal U$ is a countable subcollection of $\mathcal O$, there is some $\alpha\in\omega_1$ such that +$\bigcup\mathcal U\subseteq[0,\alpha]$, which is not dense in $X$. From 0313eeaffd5ec16f6c515d2300d6babf00798a3b Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Wed, 7 Jan 2026 00:31:40 +0100 Subject: [PATCH 10/10] Update spaces/S000035/properties/P000082.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- spaces/S000035/properties/P000082.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000035/properties/P000082.md b/spaces/S000035/properties/P000082.md index 2255c05f1..df6722ef8 100644 --- a/spaces/S000035/properties/P000082.md +++ b/spaces/S000035/properties/P000082.md @@ -4,4 +4,4 @@ property: P000082 value: true --- -For $alpha \in X$ consider the open neighborhood $U_\alpha := [0,\alpha+1)\subseteq X$. By the definition of $X$, $U_\alpha$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable). +For $\alpha \in X$ the open neighborhood $[0,\alpha + 1)\subseteq X$ is {P57} and {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable).