Arens-Fort Space is Paracompact/Proof 1
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Theorem
Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a paracompact space.
Proof
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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $26$. Arens-Fort Space: $5$