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?



A Property of Baire Spaces
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](https://math.stackexchange.com/a/88536/94514)). If $X$ has property A then $X\_1$ has empty interior and therefore $X$ is a Baire space.


