This question was inspired by [this question](https://mathoverflow.net/questions/65340/paracompact-hausdorff-but-not-compactly-generated).


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:



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


and here is the main question:



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


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.



Characterisation of paracompact spaces by some sort of embeddability?
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 $R$).


