These are the completely ultrametrizable spaces.

Recall that a d:E^{2}→[0,∞) is an ultrametric if

- d(x,y) = 0 ↔ x = y
- d(x,y) = d(y,x)
- d(x,z) ≤ max(d(x,y),d(y,z))

As usual, (E,d) is a complete ultrametric space if every Cauchy sequence converges.

Suppose E_{∞} is the inverse limit of the sequence E_{n} of discrete spaces, with f_{n}:E_{∞}→E_{n} being the limit maps. Then

d_{∞}(x,y) = inf { 2^{-n} : f_{n}(x) = f_{n}(y) }

is a complete ultrametric on E_{∞}, which is compatible with the inverse limit topology.

Conversely, given a complete ultrametric space (E,d), the relation x ∼_{n} y defined by d(x,y) ≤ 2^{-n} is an equivalence relation. Let E_{n} be the quotient E/∼_{n}, with the discrete topology. These spaces have obvious commuting maps between them, let E_{∞} be the inverse limit of this system. The map which sends each point of E to the sequence of its ∼_{n} equivalence classes is a continuous map f:E→E_{∞}. Because E is complete, this map f is a bijection. Moreover, a simple computation shows that this bijection is in fact a homeomorphism. Indeed, with d_{∞} defined as above, we have

d_{∞}(f(x),f(y)) ≥ d(x,y) ≥ d_{∞}(f(x),f(y))/2.

As Pete Clark pointed out in the comments, the above is an incomplete answer since the question does not assume that the inverse system is countable. However, the general case does admit a similar characterization in terms of uniformities. For the purposes of this answer, let us say that an *ultrauniformity* is a unformity with a fundamental system of entourages which consists of open (hence clopen) equivalence relations. The spaces in question are precisely the complete Hausdorff ultrauniform spaces.

Suppose E is the inverse limit of the discrete spaces E_{i} with limit maps f_{i}:E→E_{i}. Without loss of generality, this is a directed system. Then the sets U_{i} = {(x,y): f_{i}(x) = f_{i}(y)} form a fundamental system of entourages for the topology on E, each of which is a clopen equivalence relation on E. The universal property of inverse limits guarantees that E is complete and Hausdorff. Indeed, every Cauchy filter on E defines a compatible sequence of points in the spaces E_{i}, which is the unique limit of this filter.

Conversely, suppose E is a complete Hausdorff ultrauniform space. If U is a fundamental entourage (so U is a clopen equivalence relation on E) then the quotient space E/U is a discrete space since the diagonal is clopen. In fact, E is the inverse limit of this directed system of quotients. It is a good exercise (for Pete's students) to show that completeness and Hausdorffness of E ensure that E satisfies the universal property of inverse limits.

if infinite! Sorry. $\endgroup$ – Pete L. Clark Dec 15 '18 at 0:03