Scattered sequentially discrete spaces are anticompact

properties/P000130.md

@@ -12,4 +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.

theorems/T000813.md

@@ -0,0 +1,20 @@
+---
+uid: T000813
+if:
+  and:
+  - P000167: true
+  - P000051: true
+  - P000170: true
+then:
+  P000136: true
+---
+
+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 $x \in K' \setminus K''$).
+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).