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Let $X$ be a topological space. Are there more open sets or more closed sets? I think that there are as many open sets as closed ones.

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For every open set $U \subset X$, $X - U$ is a closed set. –  William Jun 8 '12 at 3:53

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You are correct; there is a bijection between the set of open subsets of $X$ and the set of closed subsets of $X$, $$U\longleftrightarrow X-U,$$ because by definition a set $S\subseteq X$ is closed if and only if $X-S$ is open. Because there is a bijection between the set of open subsets and the set of closed subsets, they have the same cardinality.

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