Characterisation of paracommpact paracompact spaces by some sort of embeddability?
Characterisation of paracommpact spaces by some sort of embeddability?
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
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:
and here is the main question:
There may be some very simple point-set topological properties of $I^\infy$ and paracompact spaces that give almost immediate 'no-go' theorems here, but I do not know what they would be.