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.


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 ?

  • $\begingroup$ Every subset is the complement of a subset. $\endgroup$ – 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.


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