Slight generalisation of the Baire category theorem?

Question

By the Baire category theorem, one cannot write a complete metric space $X$ as a countable union of closed nowhere dense subsets of $X$. Can this be generalised to say that there is no injection $f: X \hookrightarrow X$, $f(X) \subseteq \cup_{n \ge 1} X_n$, where the $X_n$ are closed and nowhere dense? It seems like not much of a jump, and that it should be true, but I can't quite get it out. Thanks for any insight.

Answer 1

Since you don’t require continuity, there is an injection of $\Bbb R$ into the middle-thirds Cantor set $C$, which is closed and nowhere dense, because $|\Bbb R|=|C|=2^\omega=\mathfrak c$. If you insist on a union of countably infinitely many closed, nowhere dense sets, just inject each $[n,n+1)$ into a Cantor set in $\left[n,n+\frac12\right]$.