Finite Complement Topology is not Metrizable

Theorem

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


Then $T$ is not a metrizable space.


Proof

We have:

Metrizable Space is Hausdorff
Finite Complement Space is not $T_2$ (Hausdorff)

$\blacksquare$


Sources

  • 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Zariski topology
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