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I posed this on August 8 on math.stackexchange, but there has been no response.

Let $X$ be a topological space. I define $X$ to have Property A provided that every closed meager subset of $X$ is nowhere dense. It is easy to see that all Baire spaces have Property A. Is the converse true?

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Yes. Any topological space $X$ can be decomposed as $X:=X_0 \cup X_1$ where $X_0$ is an open Baire subset of $X$ and $X_1$ is closed and meager (See e.g. this MSE answer). If $X$ has property A then $X_1$ has empty interior and therefore $X$ is a Baire space.

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