traits

Author: felixpernegger

spaces/S000130/properties/P000026.md

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+---
+space: S000130
+property: P000026
+value: false
+---
+
+The subset $\mathbb{Q}\subseteq X$ is countable and dense (look at the sets generating the topology).

spaces/S000130/properties/P000028.md

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+---
+space: S000130
+property: P000028
+value: true
+---
+
+Let $x \in X$.
+
+If $x \in C$, then the sets $\{x\}\cup (x- \delta, x + \delta)\setminus C$ with $\delta\in \mathbb{Q}_{>0}$ form a countable neighborhood basis.
+
+If $x\in A$, the sets $(x-\delta,x + \delta)\cap A$ with $\delta\in \mathbb{Q}_{>0}$ form a countable neighborhood basis.
+
+If $x\in B$, the sets $(x-\delta,x + \delta)\cap B$ with $\delta\in \mathbb{Q}_{>0}$ form a countable neighborhood basis.

spaces/S000130/properties/P000036.md

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+---
+space: S000130
+property: P000205
+value: true
+refs:
+- zb: "0386.54001"
+  name: Counterexamples in Topology
+---
+
+$X$ is {P36}, see item #3 for space #133 in {{zb:0386.54001}}.
+Furthermore $X$ refines {S25} and {S25|P205}; together this implies the assertion.

spaces/S000130/properties/P000205.md

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+---
+space: S000130
+property: P000227
+value: true
+---
+
+The subset $C\subseteq X$ is uncountable, closed (since $X \setminus C = A \cup B$) and discrete.