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- Compactness (mathematics)

Compact embedding

In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology a

K-cell (mathematics)

A -cell is a higher-dimensional version of a rectangle or rectangular solid. It is the Cartesian product of closed intervals on the real line. This means that a -dimensional rectangular solid has each

Mesocompact space

In mathematics, in the field of general topology, a topological space is said to be mesocompact if every open cover has a compact-finite open refinement. That is, given any open cover, we can find an

Σ-compact space

In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact and (weakly) locally

H-closed space

In mathematics, a Hausdorff space is said to be H-closed, or Hausdorff closed, or absolutely closed if it is closed in every Hausdorff space containing it as a subspace. This property is a generalizat

Realcompact space

In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regular Hausdorff and every point of its Stone–Čech compactification is real (meaning that t

Orthocompact space

In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open cover has an interior-preserving open refinement. That is, given an open cover of the top

Feebly compact space

In mathematics, a topological space is feebly compact if every locally finite cover by nonempty open sets is finite. The concept was introduced by S. Mardeĉić and P. Papić in 1955. Some facts:
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Lindelöf space

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which

Pseudocompact space

In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the spac

Supercompact space

In mathematics, in the field of topology, a topological space is called supercompact if there is a subbasis such that every open cover of the topological space from elements of the subbasis has a subc

Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More

A-paracompact space

In mathematics, in the field of topology, a topological space is said to be a-paracompact if every open cover of the space has a locally finite refinement. In contrast to the definition of paracompact

Sequentially compact space

In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in . Every metric space is naturally a topological spa

Exhaustion by compact sets

In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space is a nested sequence of compact subsets of (i.e. ), such that is contained in the interio

Totally bounded space

In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered b

Cocompact embedding

In mathematics, cocompact embeddings are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness. Cocompactness has been in use in mathematical analysis

Strictly singular operator

In functional analysis, a branch of mathematics, a strictly singular operator is a bounded linear operator between normed spaces which is not bounded below on any infinite-dimensional subspace.

Metacompact space

In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological sp

Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space hav

Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator , where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets

Cellular space

A cellular space is a Hausdorff space that has the structure of a CW complex.
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Limit point compact

In mathematics, a topological space X is said to be limit point compact or weakly countably compact if every infinite subset of X has a limit point in X. This property generalizes a property of compac

Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact

Hemicompact space

In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set

Relatively compact subspace

In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact.

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