Glossary of Topology - U

U

Ultra-connected

A space is ultra-connected if no two non-empty closed sets are disjoint. Every ultra-connected space is path-connected.

Ultrametric

A metric is an ultrametric if it satisfies the following stronger version of the triangle inequality: for all x, y, z in M, d(x, z) ≤ max(d(x, y), d(y, z)).

Uniform isomorphism

If X and Y are uniform spaces, a uniform isomorphism from X to Y is a bijective function f : X → Y such that f and f−1 are uniformly continuous. The spaces are then said to be uniformly isomorphic and share the same uniform properties.

Uniformizable/Uniformisable

A space is uniformizable if it is homeomorphic to a uniform space.

Uniform space

A uniform space is a set U equipped with a nonempty collection Φ of subsets of the Cartesian product X × X satisfying the following axioms:

1. if U is in Φ, then U contains { (x, x) | x in X }.
2. if U is in Φ, then { (y, x) | (x, y) in U } is also in Φ
3. if U is in Φ and V is a subset of X × X which contains U, then V is in Φ
4. if U and V are in Φ, then U ∩ V is in Φ
5. if U is in Φ, then there exists V in Φ such that, whenever (x, y) and (y, z) are in V, then (x, z) is in U.

The elements of Φ are called entourages, and Φ itself is called a uniform structure on U.

Uniform structure

See Uniform space.