Question 1

(a) Give an example of a space where the discrete topology is the same as the finite complement topology.

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- Jan 7th 2009, 04:00 PMflamingTopology Question
Question 1

(a) Give an example of a space where the discrete topology is the same as the finite complement topology. - Jan 7th 2009, 07:25 PMThePerfectHacker
Let $\displaystyle X$ be a finite set. Then the discrete topology is simply $\displaystyle \mathcal{P}(X)$. But since $\displaystyle X$ is finite it means the complement of any $\displaystyle Y\subseteq X$ is finite. Thus, the finite complement topology consists of all subsets of $\displaystyle X$ i.e. $\displaystyle \mathcal{P}(X)$.