Arens-Fort Space is Paracompact/Proof 1

From ProofWiki
Jump to navigation Jump to search


Let $T = \struct {S, \tau}$ be the Arens-Fort space.

Then $T$ is a paracompact space.


Let $\CC$ be any open cover of $T$.

Let $H \in \CC$ be any open set which contains $\tuple {0, 0}$.

For all $s \in S$ such that $s \ne \tuple {0, 0}$, we have that $\set s$ is open in $T$ by definition of the Arens-Fort space.

So the open cover of $T$ which consists of $H$ together with all the open sets $\set s$, where $s \in S \setminus H$ is a refinement of $T$ which is locally finite.

Hence the result, by definition of paracompact space.