Countable Complement Space is not Sigma-Compact

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Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$.

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


From Compact Sets in Countable Complement Space, the only compact sets in $T$ are finite.

A countable union of finite sets can not be an uncountable set.

Hence the result by definition of $\sigma$-compact space.