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.
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.