Tessellate S.T.E.M.S (2019) - Mathematics - Category C - Set 2 - Objective Problem 1

A set is closed if and only if its complement is open. Consider $X=\bigcup\limits_{n=1}^\infty S_n$ . Since $n\in S_n$ and $0\not\in S_n$ for all $n\in \mathbb{N}\setminus\{0\}$ , $X=\mathbb{N}\setminus\{0\}$ . Since $X$ is the union of open sets, $X$ itself is open and $X^C = \{0\}$ must be closed. Thus there exists a closed set of size $1$ . Since a set of size $0$ would be the empty set (and the question specifies that we are looking for a non-empty closed set), $\boxed{1}$ is the smallest possible size of a closed set.