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Page "Compact space" ¶ 87
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Countably and compact
* Countably compact spaces are pseudocompact and weakly countably compact.

Countably and finite
# REDIRECT Locally finite collection # Countably local finite collections

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 compact metric space is separable.
* 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.
* Sequentially compact: Every sequence has a convergent subsequence.
* Limit point compact: Every infinite subset has an accumulation point.
Every compact metric space is complete, though complete spaces need not be compact.
Every entire function can be represented as a power series that converges uniformly on compact sets.
* Every compact metric space ( or metrizable space ) is separable.
* Every locally compact regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.
Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem.
Every continuous function on a compact set is uniformly continuous.
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 ).
*( BCT2 ) Every locally compact Hausdorff space is a Baire space.
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.
*( BCT2 ) Every locally compact Hausdorff space is a Baire space.
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.
* σ-compact space: there exists a countable cover by compact spaces
* Every compact space is σ-compact, and every σ-compact space is Lindelöf ( i. e. every open cover has a countable subcover ).
In fact, the countable complement topology is Lindelöf but neither σ-compact nor locally compact.
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.
Formally, a topological space X is called compact if each of its open covers has a finite subcover.
Slightly more generally, any space with a finite topology ( only finitely many open sets ) is compact ; this includes in particular the trivial topology.
The open interval is not compact: the open cover for does not have a finite subcover.
* 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.

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