TITLE: Gluing of manifolds and the Hausdorff condition.
QUESTION [8 upvotes]: Hi!
I apologize in advance if this question is better fit for https://math.stackexchange.com/.
Out of curiosity I'm interested in a particular case of the problem of what properties of a manifold is given by combinatorial information associated to the gluing of the manifold from pieces of $\mathbb{R}^n$.
Let $X$ be a manifold and $K$ be a covering of $X$ by subsets of $X$ homeomorphic to $\mathbb{R}^n$. We may produce a simplicial object $N_\cdot K$ called the Čech  nerve of $K$. If the Hausdorff condition on $X$ is a constraint on the allowable gluings of $X$ from pieces of $\mathbb{R}^n$, we might expect to find this as a constraint on the set of nerves $N_\cdot K$ arising from manifolds (contra non-Hausdorff "premanifolds").
So my question becomes: Does the Hausdorff condition affect the set of possible nerves arising from such coverings? If so by what criterion? Is this criterion both necessary and sufficient? I would be interested in a characterization of such "separatedness" of simplicial objects.
Sincerely,
Eivind

REPLY [3 votes]: I am leaving this as an answer since I am new here and therefore can't comment on answers. It's not an answer, since I am not saying anything about the nerve of the covering. Apologies to the regulars here!
Here's what I wanted to say: David Carchedi's answer to the question is rather nice. However, it can be stated in much simpler terms, as follows. 
Notice that the condition on the map $\coprod U_{\alpha\beta}\to\coprod U_\alpha\times\coprod U_\beta$ to be proper is simply that it be closed, since it is injective and its domain is Hausdorff. Moreover, it's open, so it's a homeomorphism onto its image, so it is closed if and only if its range is closed. The range of the map is just the graph of the equivalence relation $R$ on $X':=\coprod U_\alpha$ such that $X'/R$ is the glued space $X$. 
Thus the statement is that $X$ is Hausdorff if and only $R$ has a closed graph in $X'\times X'$. This is true for any equivalence relation $R$ such that the canonical projection $X'\to X'/R=X$ is open. You may find this in Chapter 1, § 8.3 of Bourbaki, General Topology, vol. 1. The type of equivalence relation $R$ coming from such gluings as considered always has this property. This is also in Bourbaki, somewhere in § 4. Anyway, these things are not difficult to check by hand.