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In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. For example, in R, the closed unit interval 1 is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed).

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Wikipedia about compact

In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. For example, in R, the closed unit interval 1 is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed).

A more modern approach is to call a topological space compact if each of its open covers has a finite subcover. The Heine–Borel theorem shows that this definition is equivalent to "closed and bounded" for subsets of Euclidean space. Note: Some authors such as Bourbaki use the term "quasi-compact" instead, and reserve the term "compact" for topological spaces that are Hausdorff and "quasi-compact".

A single compact set is sometimes referred to as a compactum; following the Latin second declension (neuter), the corresponding plural form is compacta.

History and motivation

The term compact was introduced by Fréchet in 1906.

It has long been recognized that a property like compactness is necessary to prove many useful theorems. It used to be that "compact" meant "sequentially compact" (every sequence has a convergent subsequence). This was when primarily metric spaces were studied. The "covering compact" definition has become more prominent because it allows us to consider general topological spaces, and many of the old results about metric spaces can be generalized to this setting. This generalization is particularly useful in the study of function spaces, many of which are not metric spaces.

One of the main reasons for studying compact spaces is because they are in some ways very similar to finite sets: there are many results which are easy to show for finite sets, whose proofs carry over with minimal change to compact spaces. It is often said that "compactness is the next best thing to finiteness". Here is an example:

  • Suppose X is a Hausdorff space, and we have a point x in X and a finite subset A of X not containing x. Then we can separate x and A by neighbourhoods: for each a in A, let U(x) and V(a) be disjoint neighbourhoods containing x and a, respectively. Then the intersection of all the U(x) and the union of all the V(a) are the required neighbourhoods of x and A.

Note that if A is infinite, the proof fails, because the intersection of arbitrarily many neighbourhoods of x might not be a neighbourhood of x. The proof can be "rescued", however, if A is compact: we simply take a finite subcover of the cover {V(a) : a in A} of A, then intersect the corresponding finitely many U(x). In this way, we see that in a Hausdorff space, any point can be separated by neighbourhoods from any compact set not containing it. In fact, repeating the argument shows that any two disjoint compact sets in a Hausdorff space can be separated by neighbourhoods -- note that this is precisely what we get if we replace "point" (i.e. singleton set) with "compact set" in the Hausdorff separation axiom. Many of the arguments and results involving compact spaces follow such a pattern.

Search results for Compact

Compact car - Wikipedia, the free encyclopedia

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News about Compact

Audi A1 Sportback Concept - Compact Five-Door Model With Hybrid Drive

The Auto ChannelPARIS - October 1, 2008: At the Paris Motor Show (October 2-10, 2008), Audi is unveiling the A1 Sportback concept study: following on from the Audi A1 project quattro, the three-door show car presented in 2007, this is a concept for a five-door four-seater for the sub-compact class – once again exhibiting all the characteristics of a genuine Audi.http://www.theautochannel.com/F/news/2008/10/01/144763.html

HP dv3500t Compact Notebook Offers Dedicated Graphics for Under a Grand

MobileMagThere seems to be a significant price difference between equivalently-equipped compact notebooks and their slightly larger brethren. Generally speaking, it can sometimes be tough to find a smaller laptop with dedicated graphics for under a grand, but that's exactly the case with the new HP dv3500t.http://www.mobilemag.com/content/100/334/C16299/

Group bringing compact fluorescent lights to Danville homeowners

The Reidsville ReviewTeaming up with the League of Conservation Voters, the Virginia Organizing Project is delivering savings to some Danville residents’ doors.http://www.godanriver.com/gdr/news/local/danville_news/artic...

SUSSEX: Recycle your compact fluorescent bulbs

The Daily TimesDOVER — The Delaware Energy Office is launching a statewide recycling program today for compact fluorescent light bulbs.http://www.delmarvanow.com/apps/pbcs.dll/article?AID=/200810...

Ringtone royalty rate gets big boost

The TennesseanSongwriters and music publishers say they're pleased with a U.S. Copyright Royalty Board ruling issued Thursday that sets a higher rate for ringtones but keeps a 9.1-cents mechanical royalty on all other music, including compact discs and digital download services.http://www.tennessean.com/apps/pbcs.dll/article?AID=/2008100...

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Compact protects Great Lakes waters

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NRC delays decision on waste import application

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Monument, Capitol Could Get Company In DC Skyline

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HEELS ON WHEELS: 2009 VW TIGUAN SE 4MOTION REVIEW

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Welcome to Compact Data. You have reached the home page of Compact Data, a computer systems integrator with a specialty in storage systems. Feel free to email us.http://www.compact.com/
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Answers for Compact

John

Why does a compact fluorescent light bulb not work with a ceiling fan?

JohnI had this happen once with a fan that wobbled! Balanced the fan blades and the flickering quit. Btw; Does this fan have a dimmer? CFL bulbs don't like dimmers.
ivan8r

Can the use of compact fluorescent bulbs damage a home electrical system?

ivan8rtry the website energystar.gov. These lights are safe as they produce less heat then normal bulbs. They are far more energy efficient then regular bulbs. The only drawback I could find on these bul...
2fine4u

How do you install a compact washing machine?

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pudinzi

How to make minced meat more compact?

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Salad!

What is a good brand for compact refrigerators?

Salad!The only compact refrig. I've ever owned is by Haier. I've had the 1.7 cu. ft. one for a few months now and think it works just great. But, I'd really recommend buying the next size up because this ...

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