TITLE: Which spaces are inverse limits of discrete spaces ?
QUESTION [29 upvotes]: There is the following theorem:
"A space $X$ is the inverse limit of a system of discrete finite spaces, if and only if $X$ is totally disconnected, compact and Hausdorff."
A finite discrete space is totally disconnected, compact and Hausdorff and all those properties pass to inverse limits. I guess the other direction might be proved by taking the system of all decompositions of $X$ into disjoint clopen sets. The inverse limit should give $X$ back.
So what happens, if I dismiss the finiteness condition. As mentioned above every inverse limit of discrete spaces is totally disconnected, Hausdorff. So the question is:
"Which totally disconnected Hausdorff spaces are inverse limits of discrete spaces?"
For example I think it is impossible to write $\mathbb{Q}$ as an inverse limit of discrete spaces, but I don't have a proof.

REPLY [10 votes]: The rational numbers are not the inverse limit of a countable sequence of discrete spaces, but the rational numbers are in fact the inverse limit of an uncountable collection of discrete spaces. 
Another way to see that the rational numbers are not an inverse limit of a countable sequence of discrete spaces is to first take note that an inverse limit of a countable sequence of discrete spaces is metrizable by a complete metric. On the other hand, every completely metrizable subset of $\mathbb{R}$ is a $G_{\delta}$-set [DUG p. 307]. If $\mathbb{Q}$ were the intersection of countably many open sets $O_{n}$, then each $O_{n}$ would be dense making $\mathbb{Q}$ of second category. This is a contradiction. Therefore $\mathbb{Q}$ is not completely metrizable and not the inverse limit of a sequence of discrete spaces.
The rational numbers are in fact an inverse limit of discrete spaces. First take note that $\mathbb{Q}$ is
Lindelof and regular, so $\mathbb{Q}$ is realcompact [WAL p. 41]. Another way to see that $\mathbb{Q}$ is realcompact is to take note that $\mathbb{Q}$ is paracompact and of cardinality below the first measurable cardinal. Furthermore, since $\mathbb{Q}$ is Lindelof and zero-dimensional. $\mathbb{Q}$ is strongly zero-dimensional[WAL p. 85]. Therefore since $\mathbb{Q}$ is realcompact and strongly zero-dimensional, $\mathbb{Q}$ is $\mathbb{N}$-compact[WAL p. 264]. Therefore since $\mathbb{Q}$ is $\mathbb{N}$-compact, $\mathbb{Q}$ is the inverse limit of discrete spaces[CHE].
The spaces which are inverse limits of discrete spaces are precisely the spaces with a compatible complete ultrauniformity as it was pointed out earlier. We shall call a topological space ultracomplete if it can be given a compatible complete ultrauniformity. Let $X$ be a zero-dimensional space. Then let $\mathcal{U}$ be the uniformity generated by equivalence relations $E$ such that each equivalence class in $E$ is a clopen set.  Then we shall call $\mathcal{U}$ the fine ultrauniformity on the topological space $X$.  One can show that a zero-dimensional space $X$ is ultracomplete if and only if $X$ is complete in the fine ultrauniformity.
On a different note, every inverse limit of discrete spaces is a closed subspace of a product of discrete spaces [DUG p. 429]. Furthermore, one can easily show that every closed subspace of a product of discrete spaces can be given a compatible complete ultrauniformity, and hence the closed subspaces of products of discrete spaces can be written as inverse limits of discrete spaces. Therefore the spaces representable as inverse limits of discrete spaces are the spaces representable as closed subspaces of products of discrete spaces. Furthermore, for spaces of cardinality below the first measurable cardinal, the N-compact spaces correspond with the spaces representable as inverse limits of discrete spaces.
[CHE] Chew, Kim-Peu. "N-compact Spaces as Limits of Inverse Systems of Discrete Spaces." Journal of the Australian Mathematical Society 14.04 (1972): 467.
[DUG] Dugundji, James. Topology. Boston: Allyn and Bacon, 1966.
[WAL] Walker, Russell C. The Stone-Cech Compactification. Berlin: Springer-Verlag, 1974.