Book says "In the finite complement topology on a set X, the closed sets consist of X and all finite subsets of X."
What I don't understand is why are finite subsets of X closed? I thought that $T_f$ = $\{A \subset X : A = \varnothing $ or $X – A$ is finite $\} $. If U is a subset of X that is finite, and the claim is that U is a closed set, wouldn't we have to show that its complement is open? How do we know that this is true? What if X is infinite?
Best Answer
Suppose that $U$ is a finite subset of $X$. Let $V=X\setminus U$. Then $X\setminus V=U$ is finite, so by definition $V\in T_f$, and its complement, $U$, is closed.
The cardinality of $X$ is irrelevant here, though if $X$ is finite, $T_f=\wp(X)$, and every subset of $X$ is open (and closed): $T_f$ is then the discrete topology on $X$.