Uniform Space - Topology of Uniform Spaces - Uniformizable Spaces

Uniformizable Spaces

A topological space is called uniformizable if there is a uniform structure compatible with the topology.

Every uniformizable space is a completely regular topological space. Moreover, for a uniformizable space X the following are equivalent:

• X is a Kolmogorov space
• X is a Hausdorff space
• X is a Tychonoff space
• for any compatible uniform structure, the intersection of all entourages is the diagonal {(x, x) : x in X}.

Some authors (e.g. Engelking) add this last condition directly in the definition of a uniformizable space.

The topology of a uniformizable space is always a symmetric topology; that is, the space is an R₀-space.

Conversely, each completely regular space is uniformizable. A uniformity compatible with the topology of a completely regular space X can be defined as the coarsest uniformity which makes all continuous real-valued functions on X uniformly continuous. A fundamental system of entourages for this uniformity is provided by all finite intersections of sets (f × f)-1(V), where f is a continuous real-valued function on X and V is an entourage of the uniform space R. This uniformity defines a topology, which is clearly coarser than the original topology of X; that it is also finer than the original topology (hence coincides with it) is a simple consequence of complete regularity: for any x ∈ X and a neighbourhood V of x, there is a continuous real-valued function f with f(x)=0 and equal to 1 in the complement of V.

In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff space X the set of all neighbourhoods of the diagonal in X × X form the unique uniformity compatible with the topology.

A Hausdorff uniform space is metrizable if its uniformity can be defined by a countable family of pseudometrics. Indeed, as discussed above, such a uniformity can be defined by a single pseudometric, which is necessarily a metric if the space is Hausdorff. In particular, if the topology of a vector space is Hausdorff and definable by a countable family of seminorms, it is metrizable.