S195 part 1 of missing traits

properties/P000093.md

@@ -5,7 +5,9 @@ refs:
   - mr: 574525
     name: Countably compact, locally countable T2-spaces.
 ---
-The space has a base of open sets, each with countable cardinality.
+Each point has a countable neighbourhood.
+
+Equivalently, the space has a base of open sets, each with countable cardinality.
 
 ----
 #### Meta-properties

spaces/S000195/README.md

@@ -8,4 +8,7 @@ refs:
     name: Answer to "Is there a first countable, $T_1$, weakly Lindelof, sequentially compact space which is not also compact?"
 ---
 
-The set $\omega_1$ with topology equal to the join of the cofinite topology and left ray topology in the [lattice of topologies](https://en.wikipedia.org/wiki/Lattice_of_topologies) on $\omega_1$. This topology is generated by the sets of the form $\alpha\setminus F=[0,\alpha)\setminus F$ for $\alpha<\omega_1$ and $F\subseteq\omega_1$ finite.
+The set $\omega_1$ with topology equal to the join (in the [lattice of topologies](https://en.wikipedia.org/wiki/Lattice_of_topologies) on $\omega_1$) of the cofinite topology and {S217},
+i.e., the topology generated by the union of these two topologies.
+
+A base for the topology is given by the sets $[0,\alpha]\setminus F = [0,\alpha+1)\setminus F$ with $\alpha<\omega_1$ and $F\subseteq\omega_1$ finite.

spaces/S000195/properties/P000024.md

@@ -0,0 +1,7 @@
+---
+space: S000195
+property: P000024
+value: false
+---
+
+The sets $A\cup \{\omega\}$ where $A\subseteq\omega$ is cofinite form a neighbourhood base for the point $\omega$ and $\overline{A\cup \{\omega\}} = A\cup [\omega, \omega_1)$, which is not compact since the cover by the open sets $[0, \alpha)$ for $\alpha < \omega_1$ does not have a finite, or even countable subcover. 

spaces/S000195/properties/P000084.md

@@ -0,0 +1,7 @@
+---
+space: S000195
+property: P000084
+value: false
+---
+
+If $\alpha \geq \omega+\omega$, then any neighbourhood of $\alpha$ contains point $\omega+n$ for some $n\in\omega$, and any two neighbourhoods of $\alpha$ and $\omega+n$ contain a cofinite subset of $\omega$, and so intersect. It follows that $X$ is not locally Hausdorff at any point $\alpha \geq \omega+\omega$.

spaces/S000195/properties/P000085.md

@@ -0,0 +1,9 @@
+---
+space: S000195
+property: P000085
+value: false
+---
+
+If $n\in\omega$, then $\{n\} = [0, n]\setminus [0, n)$ is clopen, and so a cozero set.
+It follows that any $A\subseteq \omega$ is a cozero set as countable union of cozero sets.
+If $A\subseteq\omega$ is infinite, then $\overline{A} = A\cup [\omega, \omega_1)$, and so the closures of any two disjoint infinite sets $A, B\subseteq\omega$ intersect.

spaces/S000195/properties/P000093.md

@@ -0,0 +1,7 @@
+---
+space: S000195
+property: P000093
+value: true
+---
+
+Each point $\alpha<\omega_1$ has $[0,\alpha]$ as countable neighbourhood.