Uniform space

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In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.

The conceptual difference between uniform and topological structures is that in a uniform space, one can formalize certain notions of relative closeness and closeness of points. In other words, ideas like "x is closer to a than y is to b" make sense in uniform spaces. By comparison, in a general topological space, given sets A,B it is meaningful to say that a point x is arbitrarily close to A (i.e., in the closure of A), or perhaps that A is a smaller neighborhood of x than B, but notions of closeness of points and relative closeness are not described well by topological structure alone.

Uniform spaces generalize metric spaces and topological groups and therefore underlie most of analysis.

Contents

Definition

There are three equivalent definitions for a uniform structure.

Entourage definition

A uniform space (X, Φ) is a set X equipped with a nonempty family Φ of subsets of the Cartesian product X × X (Φ is called the uniform structure or uniformity of X and its elements entourages (French: neighborhoods or surroundings)) that satisfies the following axioms:

If the last property is omitted we call the space quasiuniform.

One usually writes U[x]={y : (x,y)∈U}. On a graph, a typical entourage is drawn as a blob surrounding the "y=x" diagonal; the U[x]’s are then the vertical cross-sections. If (x,y) ∈ U, one says that x and y are U-close. Similarly, if all pairs of points in a subset A of X are U-close (i.e., if A × A is contained in U), A is called U-small. An entourage U is symmetric if (x,y) ∈ U precisely when (y,x) ∈ U. The first axiom states that each point is U-close to itself for each entourage U. The third axiom guarantees that being "both U-close and V-close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage U there is an entourage V which is "half as large". Finally, the last axiom states the essentially symmetric property "closeness" with respect to a uniform structure.

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