How to prove that a tychonoff space $C(X, Y)$ equipped with fine topology can be embedded in a tychonoff cube.

Question

As we know that a topological space is Tychonoff if and only if it can be embedded in a Tychonoff cube. We have that a function space $C(X, Y)$ equipped with fine topology where $X$ is a Tychonoff space and $Y$ also a (Tychonoff) metric space, is a Tychonoff space. How can we then prove that it can be embedded in a hilbert cube?

Answer 1

If $Z$ is any Tychonoff space and its weight $w(Z)$ (the minimal (infinite) size of a base for $Z$) is $\kappa$ then $Z$ embeds into the Tychonoff cube $[0,1]^\kappa$ (and the weight of that cube is also $\kappa$). This can be applied to a space of the form $C(X,Y)$, when we know the topology on it is Tychonoff. It can only be embedded into the Hilbert cube if $w(C(X,Y)) = \aleph_0$ which will depend on the specific $X,Y$ of course.