Uniformizable space

<h2 id="induced-uniformity">Induced uniformity</h2>
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 <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a> on X. This is the coarsest uniformity on X for which all such functions are <a href="/facts/Uniformly_continuous/hjvw7tII">uniformly continuous</a>. A subbase for this uniformity is given by the set of all <a href="/facts/Entourage_(topology)/GKumHlHo">entourages</a>

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 {\displaystyle D_{f,\varepsilon }=\{(x,y)\in X\times X:|f(x)-f(y)|<\varepsilon \}}

where f ∈ C(X) and ε > 0.
The uniform topology generated by the above uniformity is the <a href="/facts/Initial_topology/bNRdJM81">initial topology</a> induced by the family C(X). In general, this topology will be <a href="/facts/Coarser_topology/5Q9F2fjf">coarser</a> than the given topology on X. The two topologies will coincide if and only if X is completely regular.

<h2 id="fine-uniformity">Fine uniformity</h2>
Given a uniformizable space X there is a finest uniformity on X compatible with the topology of X called the fine uniformity or universal uniformity. A uniform space is said to be fine if it has the fine uniformity generated by its uniform topology. 
The fine uniformity is characterized by the <a href="/facts/Universal_property/5XJBYMNz">universal property</a>: any continuous function f from a fine space X to a uniform space Y is uniformly continuous. This implies that the <a href="/facts/Functor/l2u3rIBE">functor</a> F : CReg → Uni that assigns to any completely regular space X the fine uniformity on X is <a href="/facts/Left_adjoint/bJ1P7AN2">left adjoint</a> to the <a href="/facts/Forgetful_functor/onEwu6HA">forgetful functor</a> sending a uniform space to its underlying completely regular space.
Explicitly, the fine uniformity on a completely regular space X is generated by all open neighborhoods D of the diagonal in X × X (with the <a href="/facts/Product_topology/0hg02IaX">product topology</a>) such that there exists a sequence D1, D2, …
of open neighborhoods of the diagonal with D = D1 and 
 
 
 
 
 D
 
 n
 
 
 ∘
 
 D
 
 n
 
 
 ⊆
 
 D
 
 n
 −
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 {\displaystyle D_{n}\circ D_{n}\subseteq D_{n-1}}
 
.
The uniformity on a completely regular space X induced by C(X) (see the previous section) is not always the fine uniformity.

<ul><li>Willard, Stephen (1970). <a href="https://archive.org/details/generaltopology00will_0">General Topology</a>. Reading, Massachusetts: Addison-Wesley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-43479-6.</li></ul>

Uniformizable space open-in-new

Uniformizable space