**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 R_{0}-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.

Read more about this topic: Uniform Space, Topology of Uniform Spaces

### Famous quotes containing the word spaces:

“Deep down, the US, with its space, its technological refinement, its bluff good conscience, even in those *spaces* which it opens up for simulation, is the only remaining primitive society.”

—Jean Baudrillard (b. 1929)