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**Drexel28** Let $\displaystyle U\subseteq X$ where $\displaystyle \#(X)<\infty$ then $\displaystyle \#\left(X-U\right)\leqslant \#(X)<\infty$ and so $\displaystyle U$ is open. Thus, the topology is just the power set, which by definition means that it's the discrete topology.

*Remark:* In fact, the reason that the cofinite topology is important is it is the coarsest $\displaystyle T_1$ topology on any set (up to homeomorphism). That said, more generally one can show that any finite $\displaystyle T_1$ space is discrete.