Fine_uniformity
Uniformizable space
Topological space whose topology is generated by a uniform structure
In mathematics, a topological space X is uniformizable if there exists a uniform structure on X that induces the topology of X. Equivalently, X is uniformizable if and only if it is homeomorphic to a uniform space (equipped with the topology induced by the uniform structure).
This article relies largely or entirely on a single source. (June 2022) |
Any (pseudo)metrizable space is uniformizable since the (pseudo)metric uniformity induces the (pseudo)metric topology. The converse fails: There are uniformizable spaces that are not (pseudo)metrizable. However, it is true that the topology of a uniformizable space can always be induced by a family of pseudometrics; indeed, this is because any uniformity on a set X can be defined by a family of pseudometrics.
Showing that a space is uniformizable is much simpler than showing it is metrizable. In fact, uniformizability is equivalent to a common separation axiom:
- A topological space is uniformizable if and only if it is completely regular.