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In mathematics, a topological space is called compact if each of its open covers has a finite subcover. Otherwise it is called non-compact.

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Wikipedia About Compact

In mathematics, a topological space is called compact if each of its open covers has a finite subcover. Otherwise it is called non-compact.

Note: Some authors such as Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff and "quasi-compact".

The Heine–Borel theorem shows that this definition is equivalent to "closed and bounded" for subsets of Euclidean space. So 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).

The concept of a compact subset of the real numbers can be extended to compact subsets of any topological space and indeed to the concept of a compact space. A subset is compact if when endowed with the subspace topology it becomes a compact space.

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 identity of bounded closed subsets of real numbers and sets whose open covers have finite subcovers was discovered and proved in the late 19th century. See Heine–Borel theorem.

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.

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Electronics every car should have

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Wikipedia results for Compact

Compact space - Wikipedia, the free encyclopedia

In mathematics, a topological space is called compact if each of its open covers has a finite subcover. Otherwise it is called non-compact. Note: Some authors such as Bourbaki use ... http://en.wikipedia.org/wiki/Compact

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Answers for Compact

faura4Why are compact light bulbs not as bright as incandescent bulbs?faura4Depends on the application sometimes they're good sometimes you need something stronger. Being that they are fluorescent the light given off is a differant color which may seem dimmer.water_skipperWhat compact digital camera takes the longest shot?water_skipperRicoh GR series have up to 180 second exposure settings.FlucsWhat is the best compact camera at the moment, in your opinion?FlucsI would say the Canon G10. The reasons are that it can be fully manual, the ability to shoot RAW, build quality, and it has a viewfinder.The One and Only OneCan a compact fluorescent lamp get wood on fire?The One and Only OneThose compact lamps develop very little heat, but it can build up over time if not vented. As long as there is some air flow, you should be all right. A few 1/2" holes in the top of your fixture...Liberty or DeathI put compact florescent bulbs in my ceiling fan and my smoke alarms began to beep randomly. Any ideas why?Liberty or DeathGenerally it is not recommended to use CFLs in vibrating environments. Vibration can cause the electronics in the CFL to fail. There is one CFL bulb (FLE11) that is available for use in a ceiling fan....
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