Can a finite set have a topology with an infinite number of open sets? ..(1)
The question originated when my professor gave us as an example that if $X$ is finite or $\tau$ is finite, $(X, \tau)$ is compact
And that that was so, even if, in the case of finite $X$ , $\tau$ had an infinite number of open sets…
Now if the topology has a finite number of open sets it is clear we can always extract a finite subcover, which is the initial cover itself, but what about the infinite case, why is it true, provided (1) is possible?
Best Answer
No. Consider that any topology $\tau$ on a set $X$ will be a subset of the powerset, that is $\tau \subseteq P(X).$ Since $X$ is finite, then $P(X)$ is finite, and consequently so is $\tau.$