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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'' 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.'' ==Induced uniformity== One way to construct a uniform structure on a topological space ''X'' is to take the initial uniformity on ''X'' induced by ''C''(''X''), the family of real-valued continuous functions on ''X''. This is the coarsest uniformity on ''X'' for which all such functions are uniformly continuous. A subbase for this uniformity is given by the set of all entourages : where ''f'' ∈ ''C''(''X'') and ε > 0. The uniform topology generated by the above uniformity is the initial topology induced by the family ''C''(''X''). In general, this topology will be coarser than the given topology on ''X''. The two topologies will coincide if and only if ''X'' is completely regular. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Uniformizable space」の詳細全文を読む スポンサード リンク
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