$X$ Polish space. Homeomorphism between $X$ and subset of $[0,1]^{\mathbb{N}}$

Question

Let $X$ be a Polish space. I already construct a homeomorphism between $X$ and a subset of $[0,1]^{\mathbb{N}}$ by: Let $\{x_1,x_2,\dots\}$ be a dense subset of $X$ and $x \mapsto (\min(1,d_X(x,x_n)))_n$. I showed that f is uniformly continuous, $f:X \to f(X)$ is bijective and that $f^{-1}:f(X) \to X$ is continuous (could'nt show uniformly for $f^{-1}$). Now I want to follow that $f(X)\subset [0,1]^{\mathbb{N}}$ is a complete subset. Any ideas?

Answer 1

Well, if $f: X \to [0,1]^{\mathbb{N}}$ is an embedding (i.e. $f:X \to f[X]$ is a homeomorphism) you cannot really do much anymore.

$f[X]$ will not be complete (in the metric inherited from the Hilbert/Tychonoff cube) in general:

$f[X]$ is complete (in said metric) iff $f[X]$ is closed in $[0,1]^{\mathbb{N}}$ iff $f[X]$ is compact iff $X$ is compact.

If you know that $X$ is Polish, i.e. completely metrisable, a classical theorem tells us that $f[X]$ is a $G_\delta$ in $[0,1]^{\mathbb{N}}$. That's all you can say about $f[X]$ in general.