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Page "Compact space" ¶ 93
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Countably and compact
* Countably compact: Every countable open cover has a finite subcover.

compact and spaces
Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.
** Tychonoff's theorem stating that every product of compact topological spaces is compact.
The hard outer layer of bones is composed of compact bone tissue, so-called due to its minimal gaps and spaces.
Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions.
This more subtle definition exhibits compact spaces as generalizations of finite sets.
In spaces that are compact in this latter sense, it is often possible to patch together information that holds locally — that is, in a neighborhood of each point — into corresponding statements that hold throughout the space, and many theorems are of this character.
Thus, what is known as the extreme value theorem in calculus generalizes to compact spaces.
In this fashion, one can prove many important theorems in the class of compact spaces, that do not hold in the context of non-compact ones.
Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.
Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
* Neither of the spaces in the previous two examples are locally compact but both are still Lindelöf
Stone spaces, compact totally disconnected Hausdorff spaces, form the abstract framework in which these spectra are studied.
* The product of any collection of compact spaces is compact.
* Two compact Hausdorff spaces X < sub > 1 </ sub > and X < sub > 2 </ sub > are homeomorphic if and only if their rings of continuous real-valued functions C ( X < sub > 1 </ sub >) and C ( X < sub > 2 </ sub >) are isomorphic.
* Compact spaces are countably compact.
* Sequentially compact spaces are countably compact.
It is often useful to embed topological spaces in compact spaces, because of the special properties compact spaces have.

compact and are
The capacity for making the distinctions of which diplomacy is compact, and the facility with language which can render them into validity in the eyes of other men are the leader's means for transforming the moral intuition into moral leadership.
There are two systematic compact layouts for a two-dimensional array.
The same phenomenon results in extremely fast spin of compact stars ( like white dwarfs, neutron stars and black holes ) when they are formed out of much larger and slower rotating stars ( indeed, decreasing the size of object 10 < sup > 4 </ sup > times results in increase of its angular velocity by the factor 10 < sup > 8 </ sup >).
There are also cycling specific multi-tools that combine many of these implements into a single compact device.
Some dioceses around the Mediterranean Sea which were Christianised early are rather compact, whereas dioceses in areas of rapid modern growth in Christian commitment — as in some parts of Sub-Saharan Africa, South America and the Far East — are much larger and more populous.
They are made up mostly of compact bone, with lesser amounts of marrow, located within the medullary cavity, and spongy bone.
* Short bones are roughly cube-shaped, and have only a thin layer of compact bone surrounding a spongy interior.
* Flat bones are thin and generally curved, with two parallel layers of compact bones sandwiching a layer of spongy bone.
In this compact arrangement, if a node has an index i, its children are found at indices ( for the left child ) and ( for the right ), while its parent ( if any ) is found at index ' ( assuming the root has index zero ).
Techniques such as Huffman coding are now used by computer-based algorithms to compress large data files into a more compact form for storage or transmission.
Without observational constraints, there are a number of candidates, such as a stable supersymmetric particle, a weakly interacting massive particle, an axion, and a massive compact halo object.
While an unlicensed satellite dish can often be identified easily, satellite radio receivers are much more compact and can rarely be easily identified, at least not without flagrantly violating provisions against unreasonable search and seizure in the Canadian Charter of Rights and Freedoms.
In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary.
Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can tend to the missing point without tending to any point within the space.
Lines and planes are not compact, since one can take a set of equally spaced points in any given direction without approaching any point.

compact and pseudocompact
The cocountable topology on an uncountable set is hyperconnected, thus connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact, hence not compact.

compact and weakly
One may call subsets of a topological vector space weakly closed ( respectively, weakly compact, etc.
In 1991, with the rediscovery of the magnetorotational instability ( MRI ), S. A. Balbus and J. F. Hawley established that a weakly magnetized disc accreting around a heavy, compact central object would be highly unstable, providing a direct mechanism for angular-momentum redistribution.
Π-indescribable cardinals are the same as weakly compact cardinals.
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence can not be proven from the standard axioms of set theory.
Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: < sup > 2 </ sup > →
A cardinal k is κ-strongly unfoldable, and κ-unfoldable, if and only if it is weakly compact.
This is not an axiom of infinity in the usual sense ; if Infinity does not hold, the closure of exists and has itself as its sole additional member ( it is certainly infinite ); the point of this axiom is that contains no additional elements at all, which boosts the theory from the strength of second order arithmetic to the strength of Morse – Kelley set theory with the proper class ordinal a weakly compact cardinal.
It in fact interprets a stronger theory ( Morse-Kelley set theory with the proper class ordinal a weakly compact cardinal ).
The theorem shows that the coarsest dual topology is the weak topology, the topology of uniform convergence on all finite subsets of, and the finest topology is the Mackey topology, the topology of uniform convergence on all weakly compact subsets of.
Given a dual pair with a locally convex space and its continuous dual then is a dual topology on if and only if it is a topology of uniform convergence on a family of absolutely convex and weakly compact subsets of
Given a dual pair with a topological vector space and its continuous dual the Mackey topology is a polar topology defined on by using the set of all absolutely convex and weakly compact sets in.
A cardinal is weakly compact if and only if it is κ-compact ; this was the original definition of that concept.
* Since every closed and bounded set is weakly relatively compact ( its closure in the weak topology is compact ), every bounded sequence in a Hilbert space H contains a weakly convergent subsequence.

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