What is uniform space?

Uniform Space

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:

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Some articles on uniform space:

Compact-open Topology - Properties
... If * is a one-point space then one can identify C(*,X) with X, and under this identification the compact-open topology agrees with the topology on X If Y is T0, T1 ... If Y is a uniform space (in particular, if Y is a metric space), then the compact-open topology is equal to the topology of compact convergence ... In other words, if Y is a uniform space, then a sequence {ƒn} converges to ƒ in the compact-open topology if and only if for every compact subset K of X, {ƒn} converges uniformly to ...
Uniform Space - History
... Before André Weil gave the first explicit definition of a uniform structure in 1937, uniform concepts, like completeness, were discussed using metric spaces ... Weil also characterized uniform spaces in terms of a family of pseudometrics ...
Filter (mathematics) - Filter On A Set - Filters in Topology - Cauchy Filters
... Let be a metric space ... Take (xn) to be a sequence in metric space X ... More generally, given a uniform space X, a filter F on X is called Cauchy filter if for every entourage U there is an A ∈ F with (x,y) ∈ U for all x,y ∈ A ...
Totally Bounded Space - Relationships With Compactness and Completeness
... Every compact metric space is totally bounded ... A uniform space is compact if and only if it is both totally bounded and Cauchy complete ... of the Heine–Borel theorem from Euclidean spaces to arbitrary spaces we must replace boundedness with total boundedness (and also replace closedness with completeness) ...

Famous quotes containing the words space and/or uniform:

    The limitless future of childhood shrinks to realistic proportions, to one of limited chances and goals; but, by the same token, the mastery of time and space and the conquest of helplessness afford a hitherto unknown promise of self- realization. This is the human condition of adolescence.
    Peter Blos (20th century)

    When a uniform exercise of kindness to prisoners on our part has been returned by as uniform severity on the part of our enemies, you must excuse me for saying it is high time, by other lessons, to teach respect to the dictates of humanity; in such a case, retaliation becomes an act of benevolence.
    Thomas Jefferson (1743–1826)