[permalink] [id link]

* Countably compact: Every countable open cover has a finite subcover.

from
Wikipedia

## Some Related Sentences

Countably and compact

Countably and finite

compact and Every

*****

__Every__topological space X is

**a**dense subspace of

**a**

__compact__space having at most one point more than X, by the Alexandroff one-point compactification

**.**

*****

__Every__continuous map from

**a**

__compact__space to

**a**Hausdorff space is closed and proper ( i

**.**e., the pre-image of

**a**

__compact__set is

__compact__

**.**

__Every__entire function can be represented as

**a**power series that converges uniformly on

__compact__sets

**.**

*****

__Every__locally

__compact__regular space is completely regular, and therefore every locally

__compact__Hausdorff space is Tychonoff

**.**

__Every__

__compact__Hausdorff space is also locally

__compact__, and many examples of

__compact__spaces may be found in the article

__compact__space

**.**

*****

__Every__

__compact__Hausdorff space of weight at most ( see Aleph number ) is the continuous image of ( this does not need the continuum hypothesis, but is less interesting in its absence ).

__Every__group

**has**

**a**presentation, and in fact many different presentations ;

**a**presentation is often the most

__compact__way of describing the structure of the group

**.**

__Every__H

*****is very special in structure

**:**it is pure-injective ( also called algebraically

__compact__), which says more or less that solving equations in H

*****is relatively straightforward

**.**

compact and countable

The last fact comes from the fact that is

__compact__Hausdorff, and hence ( since__compact__metrisable spaces are necessarily second__countable__); as well as the fact that__compact__Hausdorff spaces are metrisable exactly in case they are second__countable__**.**
An only slightly more elaborate " diagonalization " argument establishes the sequential compactness of

**a**__countable__product of sequentially__compact__spaces**.**
The operator A below can be seen to have

**a**__compact__inverse, meaning that the corresponding differential equation A f = g is solved by some integral, therefore__compact__, operator G**.**The__compact__symmetric operator G then**has****a**__countable__family of eigenvectors which are complete in**.**
Menger showed, in the 1926 construction, that the sponge is

**a**universal curve, in that any possible one-dimensional curve is homeomorphic to**a**subset of the Menger sponge, where here**a**curve means any__compact__metric space of Lebesgue covering dimension one ; this includes trees and graphs with an arbitrary__countable__number of edges, vertices and closed loops, connected in arbitrary ways**.*******

**Every**totally disconnected

__compact__metric space is homeomorphic to

**a**subset of

**a**

__countable__product of discrete spaces

**.**

*****

**Every**

__compact__space is σ-compact, and every σ-compact space is Lindelöf ( i

**.**e

**.**every

**open**

**cover**

**has**

**a**

__countable__

**subcover**).

We cannot eliminate the Hausdorff condition ;

**a**__countable__set with the indiscrete topology is__compact__,**has**more than one point, and satisfies the property that no one point sets are**open**, but is not uncountable**.**
However, the case of

**a**__compact__operator on**a**Hilbert space ( or Banach space ) is still tractable, since the eigenvalues are at most__countable__with at most**a**single limit point λ = 0**.*******sigma-algebras, sigma-fields and sigma-finiteness in measure theory ; more generally, the symbol σ serves as

**a**shorthand for " countably ", e

**.**g

**.**

**a**σ-compact topological space is one that can be written as

**a**

__countable__union of

__compact__subsets

**.**

Then it is well known that it possesses

**a**unique, up-to-scale left-( or right -) rotation invariant ring ( borel regular in the case of second__countable__) measure ( left and right probability measure in the case of__compact__), the Haar measure**.**
contains

**a**comprehensive account of the conditions on**a**second__countable__locally__compact__group G that are equivalent to amenability**:**
In dimension 4,

__compact__manifolds can have**a**__countable__infinite number of non-diffeomorphic smooth structures**.**
Mackey's original formulation was expressed in terms of

**a**locally__compact__second__countable__( lcsc ) group G,**a**standard Borel space X and**a**Borel group action*****A locally

__compact__group

**has**

**a**Følner sequence ( with the generalized definition ) if and only if it is amenable and second

__countable__

**.**

In particular, they are locally

__compact__, locally connected, first__countable__, locally contractible, and locally metrizable**.**

compact and open

In mathematics, specifically in measure theory,

**a**Borel measure is defined as follows**:**let X be**a**locally__compact__Hausdorff space, and let be the smallest σ-algebra that contains the__open__sets of X ; this is known as the σ-algebra of Borel sets**.**
) Note that the same set of points would not have, as an accumulation point, any point of the

__open__unit interval ; hence that space cannot be__compact__**.**
The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself

**:**an__open__( or half-open ) interval of the real numbers is not__compact__**.**
However, an

__open__disk is not__compact__, because**a**sequence of points can tend to the boundary without getting arbitrarily close to any point in the interior**.**
Specifically,

**a**topological space is__compact__if, whenever**a**collection of__open__sets covers the space, some subcollection consisting only of finitely many__open__sets also covers the space**.**
Slightly more generally, any space with

**a****finite**topology ( only finitely many__open__sets ) is__compact__; this includes in particular the trivial topology**.*******The set R of all real numbers is not

__compact__as there is

**a**

**cover**of

__open__intervals that does not have

**a**

**finite**

**subcover**

**.**

For example, the real line equipped with the discrete topology is closed and bounded but not

__compact__, as the collection of all singleton points of the space is an__open__**cover**which admits no**finite****subcover****.*******If the metric space X is

__compact__and an

__open__

**cover**of X is given, then there exists

**a**number such that every subset of X of diameter < δ is contained in some member of the

**cover**

**.**

More generally,

__compact__sets can be separated by__open__sets**:**if K < sub > 1 </ sub > and K < sub > 2 </ sub > are__compact__and disjoint, there exist disjoint__open__sets U < sub > 1 </ sub > and U < sub > 2 </ sub > such that and**.**
However, MusicBrainz

**has**expanded its goals to reach beyond**a**__compact__disc metadata storehouse to become**a**structured__open__online database for music**.**
It can be shown as

**a**consequence of the above properties that μ ( U ) > 0 for every non-empty__open__subset U**.**In particular, if G is__compact__then μ ( G ) is**finite**and positive, so we can uniquely specify**a**left Haar measure on G by adding the normalization condition μ ( G ) = 1**.**
In some contexts, Borel sets are defined to be generated by the

__compact__sets of the topological space, rather than the__open__sets**.*******All

__open__or closed subsets of

**a**locally

__compact__Hausdorff space are locally

__compact__in the subspace topology

**.**

This provides several examples of locally

__compact__subsets of Euclidean spaces, such as the unit disc ( either the__open__or closed version ).*****The map from X to its image in βX is

**a**homeomorphism to an

__open__subspace if and only if X is locally

__compact__Hausdorff

**.**

0.219 seconds.