Uniformizable space
Encyclopedia
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 pseudometric
s; 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
:
s 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.
The fine uniformity is characterized by the universal property
: any continuous function f from a fine space X to a uniform space Y is uniformly continuous. This implies that the functor
F : CReg → Uni which assigns to any completely regular space X the fine uniformity on X is left adjoint to the forgetful functor
which sends 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
) such that there exists a sequence D1, D2, …
of open neighborhoods of the diagonal with D = D1 and
.
The uniformity on a completely regular space X induced by C(X) (see the previous section) is not always the fine uniformity.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
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
Pseudometric space
In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space...
)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 pseudometric
Pseudometric space
In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space...
s; 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
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...
:
- 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 functionContinuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s 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
Initial topology
In general topology and related areas of mathematics, the initial topology on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous.The subspace topology and product topology constructions are both special cases of initial...
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
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
: any continuous function f from a fine space X to a uniform space Y is uniformly continuous. This implies that the functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
F : CReg → Uni which assigns to any completely regular space X the fine uniformity on X is left adjoint to the forgetful functor
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...
which sends 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
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...
) such that there exists a sequence D1, D2, …
of open neighborhoods of the diagonal with D = D1 and
.The uniformity on a completely regular space X induced by C(X) (see the previous section) is not always the fine uniformity.