I know that a subset of a nowhere dense subset is still nowhere dense. But since being of the first category doesn't imply nowhere dense, I couldn't use that fact directly. How should I proceed here?
2026-04-13 04:38:21.1776055101
Suppose F is a subset of the first category in metric space X, and E is a subset of F. Prove that E is of the first category in X.
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$F=\cup _n F_n$ where each $F_n$ is nowhere dense. Now $E=\cup_n (E \cap F_n)$ and $E \cap F_n$ is a subset of a nowhere dense set , so it is nowhere dense.