Uncountable Discrete Space is not Sigma-Compact

Theorem

Let $T = \struct {S, \tau}$ be an uncountable discrete topological space.

Then $T$ is not $\sigma$-compact.

Proof

We have that an Uncountable Discrete Space is not Lindelöf.

But a $\sigma$-compact space is Lindelöf.

So an uncountable discrete space can not be $\sigma$-compact.

$\blacksquare$

Also see

• Countable Discrete Space is Sigma-Compact

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $3$. Uncountable Discrete Topology: $8$