## 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:

... 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 ...

... 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 ...

... 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)