- Uniformizable space
In
mathematics , atopological space "X" is uniformizable if there exists auniform structure on "X" which induces the topology of "X". Equivalently, "X" is uniformizable if and only if it ishomeomorphic to a uniform space (equipped with the topology induced by the unform 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-valuedcontinuous function s on "X". This is the coarsest uniformity on "X" for which all such functions areuniformly 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.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 : any continuous function "f" from a fine space "X" to a uniform space "Y" is uniformly continuous. This implies that thefunctor "F" : CReg → Uni which assigns to any completely regular space "X" the fine uniformity on "X" isleft adjoint to theforgetful 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 the exists a sequence "D"1, "D"2, …of open neighborhoods of the diagonal with "D" = "D"1 and .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 | publisher = Addison-Wesley | location = Reading, Massachusetts | year = 1970 | id = ISBN 0-486-43479-6 (Dover edition)
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