Homeomorphism preserves category

Question

I want to show that homeomorphism preserves category. I have shown that it's true for a set of first category, but couldn't proceed for second category. There is no information given about the space and I know that completeness(a complete metric space is second category set) is not preserved under homeomorphism. So, is the claim true for a set of second category ? If it is true, how to proceed? Thanks.

Answer 1

If $f: X \to Y$ is a homeomorphism then $N$ is nowhere dense in $X$ iff $f[N]$ is nowhere dense in $f[Y]$.

Let $A$ be second category (which is a weird notion IMHO, but OK). Then suppose $f[A]$ is not second category, so it is first category; i.e. $f[A] = \cup_n N_n$ where all $N_n$ are nowhere dense sets in $Y$. Write $N_n = f[N'_n]$ where $N'_n \subseteq X$ is nowhere dense by the above fact. Note that $A = \cup_n N'_n$ would then be first category, contradiction. So $f[A]$ is second category.