Let $P\subseteq\Bbb N\cup\{\infty\}$, with $0\in P$. Does there exist a "nice" space $X_P$ such that $X_P$ has a subspace of dimension $n$ iff $n\in P$? By nice I mean nice enough that there's no need to worry about what "dimension" means, ideally a Polish space, but separable metrizable spaces would also work.
I know that surprisingly for $P=\{0,\infty\}$ there is even a compact Polish example, but I'm not sure whether there can be such gaps in the dimension of subspaces for finite dimensional spaces, so an easier question for which I'd also like to know the answer is the following: Let $X$ be a Polish space with $\dim X=n<\infty$. Must $X$ have subspaces of all dimensions between $0$ and $n$?
No, there is such an $\{n,0\}$ example in Polish spaces for every $n$. So of dimension $n$ but a non-empty subspace can only have dimension $n$ or $0$. See van Mill's book on infinite-dimensional topology (his first) for a proof and references.