From f49d3ecf7b70aa71f83c2267c1ca341a301e6b3e Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Sun, 28 Dec 2025 10:46:06 +0800 Subject: [PATCH 1/9] Scattered sequentially discrete spaces are anticompact --- properties/P000130.md | 1 + theorems/T000813.md | 19 +++++++++++++++++++ 2 files changed, 20 insertions(+) create mode 100644 theorems/T000813.md diff --git a/properties/P000130.md b/properties/P000130.md index a108331cb..f8a2d44bf 100644 --- a/properties/P000130.md +++ b/properties/P000130.md @@ -13,3 +13,4 @@ Given as condition (3) in {{wikipedia:Locally_compact_space}}. See also the art #### Meta-properties - This property is preserved by arbitrary disjoint unions. +- This property is hereditary with respect to open sets and closed sets. diff --git a/theorems/T000813.md b/theorems/T000813.md new file mode 100644 index 000000000..34cc18dae --- /dev/null +++ b/theorems/T000813.md @@ -0,0 +1,19 @@ +--- +uid: T000813 +if: + and: + - P000167: true + - P000051: true + - P000170: true +then: + P000136: true +--- + +Let $K$ be a {P16} subset of $X$. Then $K$ is {P167}, {P51} and {P130}. + +If $K$ is {P52} then $K$ is clearly {P78}. + +Otherwise, let $x \in K$ be Cantor–Bendixson rank $1$, then there exists an {P203} {P16} neighborhood $L \subseteq K$ of $x$ such that $x$ is the only non-isolated point. + +If $L$ is not finite, let $M$ be any {P181} subspace of $L$ including $x$. +Since $M$ is closed in $L$, $M$ is {P16}. However, this is impossible [(Explore)](https://topology.pi-base.org/spaces?q=181+%2B+16+%2B+167). From 0880993b0d5280d8551cdb990c19459ee7b029c2 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Mon, 29 Dec 2025 09:57:59 +0800 Subject: [PATCH 2/9] hered. locally closed --- properties/P000130.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000130.md b/properties/P000130.md index f8a2d44bf..9074b7c2e 100644 --- a/properties/P000130.md +++ b/properties/P000130.md @@ -13,4 +13,4 @@ Given as condition (3) in {{wikipedia:Locally_compact_space}}. See also the art #### Meta-properties - This property is preserved by arbitrary disjoint unions. -- This property is hereditary with respect to open sets and closed sets. +- This property is hereditary with respect to locally closed sets (equivalently, with respect to open sets and closed sets). From e6f284e13ac6d0eb37ebb7a8c8101bcdbb481b8e Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Mon, 29 Dec 2025 10:37:04 +0800 Subject: [PATCH 3/9] comment for CBR --- theorems/T000813.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000813.md b/theorems/T000813.md index 34cc18dae..1a7b758fc 100644 --- a/theorems/T000813.md +++ b/theorems/T000813.md @@ -13,7 +13,7 @@ Let $K$ be a {P16} subset of $X$. Then $K$ is {P167}, {P51} and {P130}. If $K$ is {P52} then $K$ is clearly {P78}. -Otherwise, let $x \in K$ be Cantor–Bendixson rank $1$, then there exists an {P203} {P16} neighborhood $L \subseteq K$ of $x$ such that $x$ is the only non-isolated point. +Otherwise, let $x \in K$ be Cantor–Bendixson rank $1$ (namely $x \in K' \setminus K''$), then there exists an {P203} {P16} neighborhood $L \subseteq K$ of $x$ such that $x$ is the only non-isolated point. If $L$ is not finite, let $M$ be any {P181} subspace of $L$ including $x$. Since $M$ is closed in $L$, $M$ is {P16}. However, this is impossible [(Explore)](https://topology.pi-base.org/spaces?q=181+%2B+16+%2B+167). From fbc84e6143e36b54211026b94a5423efc97e4d58 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Mon, 29 Dec 2025 12:09:01 +0800 Subject: [PATCH 4/9] adapt from P49 --- properties/P000130.md | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/properties/P000130.md b/properties/P000130.md index 9074b7c2e..76d661a49 100644 --- a/properties/P000130.md +++ b/properties/P000130.md @@ -12,5 +12,8 @@ Given as condition (3) in {{wikipedia:Locally_compact_space}}. See also the art ---- #### Meta-properties +- This property is hereditary with respect to open sets. +- This property is hereditary with respect to closed sets. +- This property is hereditary with respect to locally closed sets (equivalent to previous two meta-properties). +A set $A \subseteq X$ is called [*locally closed*](https://en.wikipedia.org/wiki/Locally_closed_subset) if every $x \in A$ has neighbourhood $U$ with $U \cap A$ closed in $U$ (equivalently, $A$ is the intersection of an open set and a closed set). - This property is preserved by arbitrary disjoint unions. -- This property is hereditary with respect to locally closed sets (equivalently, with respect to open sets and closed sets). From be0b38c1d019221150cc7cb9497020d1fbf037f9 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Mon, 29 Dec 2025 13:16:05 +0800 Subject: [PATCH 5/9] =?UTF-8?q?Compact=20discrete=20=E2=87=92=20finite?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- theorems/T000813.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000813.md b/theorems/T000813.md index 1a7b758fc..5db12eefa 100644 --- a/theorems/T000813.md +++ b/theorems/T000813.md @@ -11,7 +11,7 @@ then: Let $K$ be a {P16} subset of $X$. Then $K$ is {P167}, {P51} and {P130}. -If $K$ is {P52} then $K$ is clearly {P78}. +If $K$ is {P52} then $K$ is {P78} [(Explore)](https://topology.pi-base.org/spaces?q=16+%2B+52+%2B+%7E78). Otherwise, let $x \in K$ be Cantor–Bendixson rank $1$ (namely $x \in K' \setminus K''$), then there exists an {P203} {P16} neighborhood $L \subseteq K$ of $x$ such that $x$ is the only non-isolated point. From 97a8037558201a125d36585560720c9fc99a15c9 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Mon, 29 Dec 2025 13:32:39 +0800 Subject: [PATCH 6/9] =?UTF-8?q?mention=20k=E2=82=81T=E2=82=82?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- theorems/T000813.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/theorems/T000813.md b/theorems/T000813.md index 5db12eefa..4f9077ee6 100644 --- a/theorems/T000813.md +++ b/theorems/T000813.md @@ -9,11 +9,12 @@ then: P000136: true --- -Let $K$ be a {P16} subset of $X$. Then $K$ is {P167}, {P51} and {P130}. +Let $K$ be a {P16} subset of $X$. Then $K$ is {P130} (since $X$ is {P170}), {P167} and {P51}. If $K$ is {P52} then $K$ is {P78} [(Explore)](https://topology.pi-base.org/spaces?q=16+%2B+52+%2B+%7E78). -Otherwise, let $x \in K$ be Cantor–Bendixson rank $1$ (namely $x \in K' \setminus K''$), then there exists an {P203} {P16} neighborhood $L \subseteq K$ of $x$ such that $x$ is the only non-isolated point. +Otherwise, let $x \in K$ be Cantor–Bendixson rank $1$ (namely $x \in K' \setminus K''$). +Since $K$ is {P130}, there exists an {P203} {P16} neighborhood $L \subseteq K$ of $x$ such that $x$ is the only non-isolated point. If $L$ is not finite, let $M$ be any {P181} subspace of $L$ including $x$. Since $M$ is closed in $L$, $M$ is {P16}. However, this is impossible [(Explore)](https://topology.pi-base.org/spaces?q=181+%2B+16+%2B+167). From df2470b5e00dc6bdec49825489deb87986ab3901 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Mon, 29 Dec 2025 14:20:56 +0800 Subject: [PATCH 7/9] Update theorems/T000813.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000813.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000813.md b/theorems/T000813.md index 4f9077ee6..8e3a12b59 100644 --- a/theorems/T000813.md +++ b/theorems/T000813.md @@ -13,7 +13,7 @@ Let $K$ be a {P16} subset of $X$. Then $K$ is {P130} (since $X$ is {P170}), {P16 If $K$ is {P52} then $K$ is {P78} [(Explore)](https://topology.pi-base.org/spaces?q=16+%2B+52+%2B+%7E78). -Otherwise, let $x \in K$ be Cantor–Bendixson rank $1$ (namely $x \in K' \setminus K''$). +Otherwise, let $x \in K$ be of Cantor–Bendixson rank $1$ (namely $x \in K' \setminus K''$). Since $K$ is {P130}, there exists an {P203} {P16} neighborhood $L \subseteq K$ of $x$ such that $x$ is the only non-isolated point. If $L$ is not finite, let $M$ be any {P181} subspace of $L$ including $x$. From 225d2e1fc638a87d545f756a227a9cad90dbdfa1 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Mon, 29 Dec 2025 19:44:57 -0500 Subject: [PATCH 8/9] P170 meta-property --- properties/P000170.md | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/properties/P000170.md b/properties/P000170.md index f23e4aaa7..8ed05a52e 100644 --- a/properties/P000170.md +++ b/properties/P000170.md @@ -23,3 +23,8 @@ Equivalently, every {P16} subspace is closed in $X$ and {P130}. The property is defined in section 2 of {{doi:10.1007/BF02194829}}, where Theorem 2.1 shows the equivalences above. Note: this property is not to be confused with other variants of k-Hausdorff (e.g. {P171}), where k-closed is defined in terms of different notions of compactly generated space or k-space. + +---- +#### Meta-properties + +- This property is hereditary. From 58b7ee1b1fbe29ef86eb42784b758d9418568236 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Tue, 30 Dec 2025 08:01:49 +0100 Subject: [PATCH 9/9] Update theorems/T000813.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000813.md | 26 +++++++++++++++++++------- 1 file changed, 19 insertions(+), 7 deletions(-) diff --git a/theorems/T000813.md b/theorems/T000813.md index 8e3a12b59..443ae64cc 100644 --- a/theorems/T000813.md +++ b/theorems/T000813.md @@ -9,12 +9,24 @@ then: P000136: true --- -Let $K$ be a {P16} subset of $X$. Then $K$ is {P130} (since $X$ is {P170}), {P167} and {P51}. +Let $K$ be a {P16} subset of $X$. Then +$K$ is {P3} (since $X$ is {P170}), +and $K$ also {P167} and {P51}. +To show that $K$ is finite, without loss of generality we can assume that $X$ itself +is {P16}, {P3}, {P167} and {P51} +and show that $X$ is finite. -If $K$ is {P52} then $K$ is {P78} [(Explore)](https://topology.pi-base.org/spaces?q=16+%2B+52+%2B+%7E78). +If every point of $X$ is isolated, $X$ is finite by compactness. +Otherwise, because $X$ is {P51}, choose $x \in X$ with Cantor–Bendixson rank $1$ (that is, $x\in X'\setminus X''$). +So $x$ is not isolated in $X$ and there is a neighbourhood $L$ of $x$ +such that all point in $L\setminus\{x\}$ are isolated in $X$. +Since $X$ is {P11}, we can assume $L$ is closed in $X$. +The neighbourhood $L$ must be infinite, +because otherwise $x$ would be isolated. -Otherwise, let $x \in K$ be of Cantor–Bendixson rank $1$ (namely $x \in K' \setminus K''$). -Since $K$ is {P130}, there exists an {P203} {P16} neighborhood $L \subseteq K$ of $x$ such that $x$ is the only non-isolated point. - -If $L$ is not finite, let $M$ be any {P181} subspace of $L$ including $x$. -Since $M$ is closed in $L$, $M$ is {P16}. However, this is impossible [(Explore)](https://topology.pi-base.org/spaces?q=181+%2B+16+%2B+167). +Take a countably infinite set $M\subseteq L$ containing $x$. +Since $L$ is {P203}, $M$ is closed in $L$, hence closed in $X$ and compact. +And since $M$ is {P167} and {P16}, +it is not {P181} +[(Explore)](https://topology.pi-base.org/spaces?q=167+%2B+16+%2B+181), +which is a contradiction.