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.
Read more about this topic: Uniformizable Space
Famous quotes containing the words induced and/or uniformity:
“It is a misfortune that necessity has induced men to accord greater license to this formidable engine, in order to obtain liberty, than can be borne with less important objects in view; for the press, like fire, is an excellent servant, but a terrible master.”
—James Fenimore Cooper (1789–1851)
“The diversity in the faculties of men, from which the rights of property originate, is not less an insuperable obstacle to a uniformity of interests. The protection of these faculties is the first object of government.”
—James Madison (1751–1836)