Arens-Fort Space is Paracompact/Proof 2

Theorem

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

Then $T$ is a paracompact space.

Proof

We have that:

The Arens-Fort Space is Completely Normal.

The Arens-Fort Space is Lindelöf.

From Sequence of Implications of Separation Axioms, it follows that $T$ is a $T_3$ space.

The result follows from Lindelöf $T_3$ Space is Paracompact.

$\blacksquare$

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: $5$