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Compact space

In [mathematics] , more specifically [general topology] and [metric topology] , a ** compact space ** is an abstract mathematical [space] in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space. Thus a [closed] and [bounded] subset (such as a [closed interval] or [rectangle] ) of a [Euclidean space] is compact because ultimately one's steps are forced to "bunch up" near a point of the set, a result known as the [Bolzano–Weierstrass theorem] , whereas Euclidean space itself is not compact because one can take infinitely many equal steps in any given direction without ever getting very close to any other point of the space.

Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of [functions] . The term * compact * was introduced into mathematics by [Maurice Fréchet] in 1906 as a distillation of this concept. Compactness in this more general situation plays an extremely important role in [mathematical analysis] , because many classical and important theorems of 19th century analysis, such as the [extreme value theorem] , are easily generalized to this situation. A typical application is furnished by the [Arzelà–Ascoli theorem] and in particular the [Peano existence theorem] , in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.

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