Excluded Point Space is Sequentially Compact

Theorem

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.

Then $T$ is a sequentially compact space.

Proof

Excluded Point Space is Compact

Compact Space is Countably Compact

Excluded Point Space is First-Countable

First-Countable Space is Sequentially Compact iff Countably Compact

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $13 \text { - } 15$. Excluded Point Topology: $6$