# Compactness (Mathematics)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 28. Chapters: Compactness theorems, Compact space, Heine-Borel theorem, Locally compact space, Paracompact space, Bolzano-Weierstrass theorem, Arzelà-Ascoli theorem, Sobolev inequality, Totally bounded space, Compact operator, Banach-Alaoglu theorem, Lindelöf space, Supercompact space, Prokhorov's theorem, Helly's selection theorem, Fra?ková-Helly selection theorem, Montel's theorem, Limit point compact, Σ-compact space, Compactly embedded, Metacompact space, Pseudocompact space, Mazur's lemma, Sequentially compact space, Hemicompact space, Orthocompact space, Eberlein-Šmulian theorem, Relatively compact subspace, Exhaustion by compact sets, Strictly singular operator, Mesocompact space, Realcompact space, Feebly compact space, A-paracompact space. Excerpt: In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces. Compactness is not easy to describe precisely in an intuitive manner; in some sense it says that the topology allows the space to be considered as "small" (compactness is a kind of topological counterpart to finiteness of sets), even though as a set it may be quite large. Moreover, more intuitive characterizations of compactness are often dependent on additional properties of the topological space to be valid; the following description assumes the space is a metric space, so that "closeness" of points has meaning. Then compactness means that whenever one chooses infinitely many sample points from the space, there is bound to be at least one point of the space to which some of the samples ultimately get arbitrarily close. This could be because some point is itself sampled infinitely many times (as would necessarily happen if the space were finite), but ...

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