# Given two topologies on a set , prove that the set is finite.

Problem

Let $\tau$ be a finite closed topology on a set X . If $\tau$ is also the discrete topology, prove that the set X is finite.

Attempt

Since $\tau$ is a discrete topology ,it contains all its subsets. Also $\tau$ is finite closed topology ,which implies all subsets in $\tau$ has finite complement . Combining these two we get, all the subsets of X have finite complement,i.e., X-S$_i$ = finite for all i.

How to proceed after that ?

• Every subset is the complement of a subset. – egreg Oct 2 '18 at 14:15

Let $$x \in X$$, then $$\{x\}$$ is open (as $$\tau$$ is discrete) and so $$X\setminus \{x\}$$ is finite. Now $$X = \{x\} \cup X\setminus \{x\}$$ which is still finite.