Consider the space $X=\{0,1\}^{\mathbf{N}}$ endowed with the product topology (of the discrete topology on $\{0,1\}$).
Question. Does there exist a second category set $S\subseteq X$ with empty interior?
2026-03-28 10:17:33.1774693053
Second category set with empty interior in $X=\{0,1\}^{\mathbf{N}}$
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$X$ is compact metrisable so second category in itself. If $D$ is countable and dense then $S=X\setminus D$ is second category and has empty interior.