Uniformizable Space

Uniformizable Space

In mathematics, a topological space X is uniformizable if there exists a uniform structure on X which 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).

Any (pseudo)metrizable space is uniformizable since the (pseudo)metric uniformity induces the (pseudo)metric topology. The converse fails: There are uniformizable spaces which are not (pseudo)metrizable. However, it is true that the topology of a uniformizable space can always be induced by a family (mathematics) 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.

Read more about Uniformizable Space:  Induced Uniformity, Fine Uniformity

Famous quotes containing the word space:

    True spoiling is nothing to do with what a child owns or with amount of attention he gets. he can have the major part of your income, living space and attention and not be spoiled, or he can have very little and be spoiled. It is not what he gets that is at issue. It is how and why he gets it. Spoiling is to do with the family balance of power.
    Penelope Leach (20th century)