Characterisation of paracompact spaces by some sort of embeddability?

Question

This question was inspired by this question.

Before I start, I don't really mean embedding in what follows. I'm tempted to use plongement, for an exotic touch, but well, that's just a rose by another name.

Consider a paracompact space $X$ with an open cover $\{U_a\}_{a\in A}$ given by a partition of unity $\phi = \{\phi_a\}_{a\in A}$ (i.e. $U_a = \phi_a^{-1}(0,1]$). Then given a total order on $A$, we can construct a map $\Phi:X \to \lim_{n\to \infty} [0,1]^n =: I^\infty$, the latter being the space of sequences valued in $[0,1]$ with at most a finite number of non-zero values. Now I have no idea what sort of properties we can assume this map has. I would like to think that with suitable manipulation of $\phi$ we could make this at least a local embedding. If this locally-embedded-ing is handled carefully, it should lead to a global embedding.

My argument is as follows: Say we take a point $x\in X$, and consider those $\phi_a$ with support at $x$. Since partitions of unity are locally finite, we can instead consider those $\phi_a$ (a finite number) which have support in an open neighbourhood of $x$: denote these by $\phi_1,\ldots,\phi_N$. Take the intersection $U_1\cap\ldots\cap U_N$ and call this open set $W$. If we can manipulate $\phi$ on $W$ (to $\phi'$, say) such that $\Phi'|_W$ is an embedding, then we should be able assume that $\Phi$ is an embedding. Perhaps one needs to go through the previous paragraph and say injective instead of embedding.

But I think that my argument is too weak and/or faulty. I see no way of ensuring that $\Phi|_W$ is an embedding/injective. But the existence of such a function $\Phi$ for any open cover (not uniquely of course - one may need to pass to a refinement) seems like a way to characterise paracompactness (in a way I hope is not a mere relabelling). For, consider a basis $B$ for the topology on $X$ (a collection of open sets from which we get all the others by arbitrary unions). Then we get a map $\Phi_B:X\to I^\infty$. From this map we get a topology on $X$. Here (finally) is a question:

Is the topology on a paracompact space $X$ induced by a map $X\to I^\infty$?

and here is the main question:

Can we (or the literature) characterise paracompact spaces in this way?

There may be some very simple point-set topological properties of $I^\infty$ and paracompact spaces that give almost immediate 'no-go' theorems here, but I do not know what they would be.

Answer 1

You cannot embed all paracompact spaces in this way. Even if we allow at most countably many non-zero coordinates: in that case we get subspaces of $\Sigma$-products, which have been well-studied in general topology and functional analysis. Compact subspaces of these are called Corson compact spaces, and none of these map onto $[0,1]^{\omega_1}$. So the latter space is an example of a compact Hausdorff space (so paracompact etc.) that cannot be embedded into a $\Sigma$-product of copies of $I$ (or $R$).