From 83e677509fea0129299c76d56702e7295f10cec2 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 17 Jan 2026 20:34:12 +0100 Subject: [PATCH 1/9] traits --- properties/P000026.md | 6 ++++++ spaces/S000207/properties/P000026.md | 7 +++++++ spaces/S000207/properties/P000028.md | 10 ++++++++++ spaces/S000207/properties/P000051.md | 12 ++++++++++++ spaces/S000207/properties/P000093.md | 7 +++++++ spaces/S000207/properties/P000114.md | 10 ++++++++++ 6 files changed, 52 insertions(+) create mode 100644 spaces/S000207/properties/P000026.md create mode 100644 spaces/S000207/properties/P000028.md create mode 100644 spaces/S000207/properties/P000051.md create mode 100644 spaces/S000207/properties/P000093.md create mode 100644 spaces/S000207/properties/P000114.md diff --git a/properties/P000026.md b/properties/P000026.md index f23fb7fa7a..f666c801dc 100644 --- a/properties/P000026.md +++ b/properties/P000026.md @@ -11,3 +11,9 @@ refs: A space with a countable dense subset. Defined on page 7 of {{zb:0386.54001}}. + +---- +#### Meta-properties + +- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does. +- This property is hereditary. diff --git a/spaces/S000207/properties/P000026.md b/spaces/S000207/properties/P000026.md new file mode 100644 index 0000000000..5289ec20cd --- /dev/null +++ b/spaces/S000207/properties/P000026.md @@ -0,0 +1,7 @@ +--- +space: S000207 +property: P000026 +value: false +--- + +$\omega_1 \times \omega_1 \subseteq X$ is discrete and uncountable and therefore clearly not separable. diff --git a/spaces/S000207/properties/P000028.md b/spaces/S000207/properties/P000028.md new file mode 100644 index 0000000000..10cebca40f --- /dev/null +++ b/spaces/S000207/properties/P000028.md @@ -0,0 +1,10 @@ +--- +space: S000207 +property: P000028 +value: false +refs: + - wikipedia: Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets + name: Cofinality on Wikipedia +--- + +Each point $\langle \alpha,\omega_1\rangle$ has a local basis of sets of the form $(\beta,\alpha]\times(\gamma,\omega_1]$ for $\beta<\alpha$ and $\gamma<\omega_1$, the assertion follows since $\omega_1$ has uncountable cofinality (see {{wikipedia:Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets}}). diff --git a/spaces/S000207/properties/P000051.md b/spaces/S000207/properties/P000051.md new file mode 100644 index 0000000000..2faeaf9222 --- /dev/null +++ b/spaces/S000207/properties/P000051.md @@ -0,0 +1,12 @@ +--- +space: S000207 +property: P000051 +value: true +--- + +Let $Y\subseteq X$. + +If $Y \cap \omega_1 \times \omega_1 \neq \emptyset$, this follows from the definition. + +Otherwise there is an $A \subseteq \omega_1$ such that $Y = A \times \{\omega_1\}$. Let $p = \inf A$, +then $\{\langle p, \omega_1 \rangle\} = Y \cap (-1,p] \times (0,\omega_1]$ is open in $Y$. diff --git a/spaces/S000207/properties/P000093.md b/spaces/S000207/properties/P000093.md new file mode 100644 index 0000000000..8c05b4b686 --- /dev/null +++ b/spaces/S000207/properties/P000093.md @@ -0,0 +1,7 @@ +--- +space: S000207 +property: P000093 +value: false +--- + +Each point $\langle \alpha,\omega_1\rangle$ has a local basis of sets of the form $(\beta,\alpha]\times(\gamma,\omega_1]$ for $\beta<\alpha$ and $\gamma<\omega_1$, all of which are uncountable. diff --git a/spaces/S000207/properties/P000114.md b/spaces/S000207/properties/P000114.md new file mode 100644 index 0000000000..d9e4e9e82e --- /dev/null +++ b/spaces/S000207/properties/P000114.md @@ -0,0 +1,10 @@ +--- +space: S000207 +property: P000114 +value: true +refs: + - mathse: 1383755 + name: Cardinality of the Cartesian Product of Two Equinumerous Infinite Sets +--- + +Since $|\omega_1|=|\omega_1 + 1| = \aleph_1$. See for example {{mathse:1383755}}. From 6125fc821b2a209acbc6823649a701519bd51538 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Sun, 18 Jan 2026 02:33:52 +0100 Subject: [PATCH 2/9] no metaproperty after all --- properties/P000026.md | 1 - 1 file changed, 1 deletion(-) diff --git a/properties/P000026.md b/properties/P000026.md index f666c801dc..de834aa8a5 100644 --- a/properties/P000026.md +++ b/properties/P000026.md @@ -16,4 +16,3 @@ Defined on page 7 of {{zb:0386.54001}}. #### Meta-properties - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does. -- This property is hereditary. From d1844b745b632308c1637b97e01d6b37e9c267ea Mon Sep 17 00:00:00 2001 From: Batixx Date: Sun, 18 Jan 2026 18:27:02 +0100 Subject: [PATCH 3/9] stregthen theorems --- spaces/S000207/properties/P000026.md | 7 ------- spaces/S000207/properties/P000062.md | 7 +++++++ spaces/S000207/properties/P000081.md | 7 +++++++ spaces/S000207/properties/P000093.md | 7 ------- 4 files changed, 14 insertions(+), 14 deletions(-) delete mode 100644 spaces/S000207/properties/P000026.md create mode 100644 spaces/S000207/properties/P000062.md create mode 100644 spaces/S000207/properties/P000081.md delete mode 100644 spaces/S000207/properties/P000093.md diff --git a/spaces/S000207/properties/P000026.md b/spaces/S000207/properties/P000026.md deleted file mode 100644 index 5289ec20cd..0000000000 --- a/spaces/S000207/properties/P000026.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000207 -property: P000026 -value: false ---- - -$\omega_1 \times \omega_1 \subseteq X$ is discrete and uncountable and therefore clearly not separable. diff --git a/spaces/S000207/properties/P000062.md b/spaces/S000207/properties/P000062.md new file mode 100644 index 0000000000..edf7b15013 --- /dev/null +++ b/spaces/S000207/properties/P000062.md @@ -0,0 +1,7 @@ +--- +space: S000207 +property: P000062 +value: false +--- + +Consider the open cover $\{[0,\alpha]\times (\omega_1 + 1) \mid \alpha \in \omega_1\}$. This has no dense countable subcollection by a similar argument given as in {S35|P62}. diff --git a/spaces/S000207/properties/P000081.md b/spaces/S000207/properties/P000081.md new file mode 100644 index 0000000000..4ead599c6c --- /dev/null +++ b/spaces/S000207/properties/P000081.md @@ -0,0 +1,7 @@ +--- +space: S000207 +property: P000081 +value: false +--- + +$X$ contains $$\{0\}\times (\omega_1 + 1)$ which is homeomorphic to {S155} and {S155|P81}. diff --git a/spaces/S000207/properties/P000093.md b/spaces/S000207/properties/P000093.md deleted file mode 100644 index 8c05b4b686..0000000000 --- a/spaces/S000207/properties/P000093.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000207 -property: P000093 -value: false ---- - -Each point $\langle \alpha,\omega_1\rangle$ has a local basis of sets of the form $(\beta,\alpha]\times(\gamma,\omega_1]$ for $\beta<\alpha$ and $\gamma<\omega_1$, all of which are uncountable. From db68e627dc39e06cf29ea9974278a40b87edea63 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Mon, 19 Jan 2026 10:28:34 +0100 Subject: [PATCH 4/9] Update spaces/S000207/properties/P000081.md Co-authored-by: yhx-12243 --- spaces/S000207/properties/P000081.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000207/properties/P000081.md b/spaces/S000207/properties/P000081.md index 4ead599c6c..f83443ed52 100644 --- a/spaces/S000207/properties/P000081.md +++ b/spaces/S000207/properties/P000081.md @@ -4,4 +4,4 @@ property: P000081 value: false --- -$X$ contains $$\{0\}\times (\omega_1 + 1)$ which is homeomorphic to {S155} and {S155|P81}. +$X$ contains $\{0\}\times (\omega_1 + 1)$ which is homeomorphic to {S155} and {S155|P81}. From 87e328f87d61c1075ab582ac0577e60f6f234c80 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 20 Jan 2026 03:05:54 +0100 Subject: [PATCH 5/9] remove redundant --- spaces/S000207/properties/P000028.md | 10 ---------- 1 file changed, 10 deletions(-) delete mode 100644 spaces/S000207/properties/P000028.md diff --git a/spaces/S000207/properties/P000028.md b/spaces/S000207/properties/P000028.md deleted file mode 100644 index 10cebca40f..0000000000 --- a/spaces/S000207/properties/P000028.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000207 -property: P000028 -value: false -refs: - - wikipedia: Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets - name: Cofinality on Wikipedia ---- - -Each point $\langle \alpha,\omega_1\rangle$ has a local basis of sets of the form $(\beta,\alpha]\times(\gamma,\omega_1]$ for $\beta<\alpha$ and $\gamma<\omega_1$, the assertion follows since $\omega_1$ has uncountable cofinality (see {{wikipedia:Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets}}). From 2894ce1914f88fc3e170b0183b185688c0b68f5e Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 20 Jan 2026 03:07:09 +0100 Subject: [PATCH 6/9] Update spaces/S000207/properties/P000081.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- spaces/S000207/properties/P000081.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000207/properties/P000081.md b/spaces/S000207/properties/P000081.md index f83443ed52..2fb60c4fe7 100644 --- a/spaces/S000207/properties/P000081.md +++ b/spaces/S000207/properties/P000081.md @@ -4,4 +4,4 @@ property: P000081 value: false --- -$X$ contains $\{0\}\times (\omega_1 + 1)$ which is homeomorphic to {S155} and {S155|P81}. +The subspace $\{0\}\times (\omega_1 + 1)\subseteq X$ is homeomorphic to {S155} and {S155|P81}. From f2aaf3f3043d5253072556d50c65931dc48df59a Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 20 Jan 2026 04:16:56 +0100 Subject: [PATCH 7/9] remove mathse reference --- spaces/S000207/properties/P000114.md | 5 +---- 1 file changed, 1 insertion(+), 4 deletions(-) diff --git a/spaces/S000207/properties/P000114.md b/spaces/S000207/properties/P000114.md index d9e4e9e82e..cbe8f47777 100644 --- a/spaces/S000207/properties/P000114.md +++ b/spaces/S000207/properties/P000114.md @@ -2,9 +2,6 @@ space: S000207 property: P000114 value: true -refs: - - mathse: 1383755 - name: Cardinality of the Cartesian Product of Two Equinumerous Infinite Sets --- -Since $|\omega_1|=|\omega_1 + 1| = \aleph_1$. See for example {{mathse:1383755}}. +Since $|\omega_1|=|\omega_1 + 1| = \aleph_1$. From 4ae9f177d95402da868f77717b1b493f1adecdb3 Mon Sep 17 00:00:00 2001 From: Batixx Date: Tue, 20 Jan 2026 09:53:19 +0100 Subject: [PATCH 8/9] update scattered --- properties/P000051.md | 4 ++++ spaces/S000207/properties/P000051.md | 7 +------ 2 files changed, 5 insertions(+), 6 deletions(-) diff --git a/properties/P000051.md b/properties/P000051.md index 53cd08d15e..6ec56ca038 100644 --- a/properties/P000051.md +++ b/properties/P000051.md @@ -25,3 +25,7 @@ Defined as "dispersed" in e.g. {{zb:0164.53101}}. #### Meta-properties - This property is hereditary. +- This property is hereditary. +- This property is preserved by finer topologies. +- This property is preserved by arbitrary disjoint unions. +- This property is preserved by finite products. diff --git a/spaces/S000207/properties/P000051.md b/spaces/S000207/properties/P000051.md index 2faeaf9222..ec18b39460 100644 --- a/spaces/S000207/properties/P000051.md +++ b/spaces/S000207/properties/P000051.md @@ -4,9 +4,4 @@ property: P000051 value: true --- -Let $Y\subseteq X$. - -If $Y \cap \omega_1 \times \omega_1 \neq \emptyset$, this follows from the definition. - -Otherwise there is an $A \subseteq \omega_1$ such that $Y = A \times \{\omega_1\}$. Let $p = \inf A$, -then $\{\langle p, \omega_1 \rangle\} = Y \cap (-1,p] \times (0,\omega_1]$ is open in $Y$. +$X$ is finer than $\omega_1 \times (\omega_1 + 1)$ with the product topology, {S35|P51} and {S36|P51}. From 7f8a9d812801495bb0e73ee383ab37e7a0ed7851 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 20 Jan 2026 12:53:34 +0100 Subject: [PATCH 9/9] typo --- properties/P000051.md | 1 - 1 file changed, 1 deletion(-) diff --git a/properties/P000051.md b/properties/P000051.md index 6ec56ca038..d61edd2c07 100644 --- a/properties/P000051.md +++ b/properties/P000051.md @@ -24,7 +24,6 @@ Defined as "dispersed" in e.g. {{zb:0164.53101}}. ---- #### Meta-properties -- This property is hereditary. - This property is hereditary. - This property is preserved by finer topologies. - This property is preserved by arbitrary disjoint unions.