In mathematics, a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

''X'' is uniformizable if

there exists In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, wh ...

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). Any (

pseudo The prefix pseudo- (from Greek ψευδής, ''pseudes'', "false") is used to mark something that superficially appears to be (or behaves like) one thing, but is something else. Subject to context, ''pseudo'' may connote coincidence, imitation, ...

)

metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty ...

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 Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...

'' of pseudometrics; indeed, this is because any uniformity on a set ''X'' can be

defined A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...

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 In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometime ...

: :''A topological space is uniformizable if and only if it is

completely regular In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that i ...

.''

Induced uniformity

One way to construct a uniform structure on a topological space ''X'' is to take the

initial uniformity In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph tha ...

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 In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. I ...

. A subbase for this uniformity is given by the set of all

entourage An entourage () is an informal group or band of people who are closely associated with a (usually) famous, notorious, or otherwise notable individual. The word can also refer to: Arts and entertainment * L'entourage, French hip hop / rap collecti ...

s :

D_ = \

where ''f'' ∈ ''C''(''X'') and ''ε'' > 0. The uniform topology generated by the above uniformity is the

initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' ...

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.

Fine uniformity

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

universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...

: any continuous function ''f'' from a fine space ''X'' to a uniform space ''Y'' is uniformly continuous. This implies that the

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

''F'' : CReg → Uni that assigns to any completely regular space ''X'' the fine uniformity on ''X'' is

left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

to the

forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sig ...

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

product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...

) such that there exists a sequence ''D''₁, ''D''₂, … of open neighborhoods of the diagonal with ''D'' = ''D''₁ and

D_n\circ D_n\subseteq D_

. The uniformity on a completely regular space ''X'' induced by ''C''(''X'') (see the previous section) is not always the fine uniformity.

References

*{{cite book , last = Willard , first = Stephen , title = General Topology , url = https://archive.org/details/generaltopology00will_0 , url-access = registration , publisher = Addison-Wesley , location = Reading, Massachusetts , year = 1970 , isbn = 0-486-43479-6 Properties of topological spaces Uniform spaces