# Dictionary Definition

compact adj

1 closely and firmly united or packed together;
"compact soil"; "compact clusters of flowers" [ant: loose]

2 closely crowded together; "a compact shopping
center"; "a dense population"; "thick crowds" [syn: dense, thick]

3 heavy and compact in form or stature; "a
wrestler of compact build"; "he was tall and heavyset"; "stocky
legs"; "a thick middle-aged man"; "a thickset young man" [syn:
heavyset, stocky, thick, thickset]

4 briefly giving the gist of something; "a short
and compendious book"; "a compact style is brief and pithy";
"succinct comparisons"; "a summary formulation of a wide-ranging
subject" [syn: compendious, succinct, summary]

### Noun

1 a small cosmetics case with a mirror; to be
carried in a woman's purse [syn: powder
compact]

2 a signed written agreement between two or more
parties (nations) to perform some action [syn: covenant, concordat]

3 a small and economical car [syn: compact
car]

### Verb

1 have the property of being packable or
compactable or of compacting easily; "This powder compacts easily";
"Such odd-shaped items do not pack well" [syn: pack]

3 make more compact by or as if by pressing;
"compress the data" [syn: compress, pack
together] [ant: decompress]

4 squeeze or press together; "she compressed her
lips"; "the spasm contracted the muscle" [syn: compress, constrict, squeeze, contract, press]

# User Contributed Dictionary

## English

# Extensive Definition

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 [0, 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.

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 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.

## Definitions

### Compactness of subsets of Rn

For any subset of Euclidean space Rn, the following four conditions are equivalent:- Every open cover has a finite subcover. This is the most commonly used definition.
- Every sequence in the set has a convergent subsequence, the limit point of which belongs to the set.
- Every infinite subset of the set has an accumulation point in the set.
- The set is closed and bounded. This is the condition that is easiest to verify, for example a closed interval or closed n-ball.

In other spaces, these
conditions may or may not be equivalent, depending on the
properties of the space.

Note that while compactness is
a property of the set itself (with its topology), closedness is
relative to a space it is in; above "closed" is used in the sense
of closed in Rn. A set which is closed in e.g. Qn is typically not
closed in Rn, hence not compact.

### Compactness of topological spaces

The "finite subcover" property from the previous paragraph is more abstract than the "closed and bounded" one, but it has the distinct advantage that it can be given using the subspace topology on a subset of Rn, eliminating the need of using a metric or an ambient space. Thus, compactness is a topological property. In a sense, the closed unit interval [0,1] is intrinsically compact, regardless of how it is embedded in R or Rn.A topological space X is
defined as compact if all its open covers have a finite subcover.
Formally, this means that

- for every arbitrary collection \_ of open subsets of X such that \bigcup_ U_\alpha \supseteq X, there is a finite subset J\subset A such that \bigcup_ U_i \supseteq X.

An often used equivalent
definition is given in terms of the
finite intersection property: if any collection of closed sets
satisfying the finite intersection property has nonempty
intersection, then the space is compact. This definition is dual to
the usual one stated in terms of open sets.

Some authors require that a
compact space also be Hausdorff,
and the non-Hausdorff version is then called
quasicompact.

## Examples of compact spaces

- Any finite topological space, including the empty set, is compact. Slightly more generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology.
- The closed unit interval [0, 1] is compact. This follows from the Heine-Borel theorem; the proof of which is about as hard as proving directly that [0,1] is compact. The open interval (0,1) is not compact: the open cover (1/n, 1-1/n) for n=3,4,... does not have a finite subcover.
- For every natural number n, the n-sphere is compact. Again from the Heine-Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.
- The Cantor set is compact. Since the p-adic integers are homeomorphic to the Cantor set, they also form a compact set. Since a finite set containing p elements is compact, this shows that the countable product of finite sets is compact, and is thus a special case of Tychonoff's theorem.
- Consider the set K of all functions f: \mathbb \rightarrow [0,1] from the real number line to the closed unit interval, and define a topology on K so that a sequence \ in K converges towards f\in K if and only if \ converges towards f(x) for all x\in\mathbb. There is only one such topology; it is called the topology of pointwise convergence. Then K is a compact topological space, again a consequence of Tychonoff's theorem.
- Consider the set K of all functions f\colon [0,1]\to [0,1] satisfying the Lipschitz condition |f(x)-f(y)|\le |x-y| for all x,y\in[0,1] and consider on K the metric induced by the uniform distance d(f,g)=\sup\. Then by Ascoli-Arzelà theorem the space K is compact.
- Any space carrying the cofinite topology is compact.
- Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification. The one-point compactification of \mathbb is homeomorphic to the circle S^1; the one-point compactification of \mathbb^2 is homeomorphic to the sphere S^2. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
- The spectrum of any continuous linear operator on a Hilbert space is a compact subset of the complex numbers C. If the Hilbert space is infinite-dimensional, then any compact subset of C arises in this manner, as the spectrum of some continuous linear operator on the Hilbert space.
- The spectrum of any commutative ring or Boolean algebra is compact.
- The Hilbert cube is compact.
- The right order topology or left order topology on any bounded totally ordered set is compact. In particular, Sierpinski space is compact.
- The prime spectrum of any commutative ring with the Zariski topology is a compact space, important in algebraic geometry. These prime spectra are almost never Hausdorff spaces.

## Theorems

Some theorems related to compactness (see the Topology Glossary for the definitions):- A continuous image of a compact space is compact.
- The extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded and attains its supremum.
- A closed subset of a compact space is compact.
- A compact subset of a Hausdorff space is closed.
- A nonempty compact subset of the real numbers has a greatest element and a least element.
- A subset of Euclidean n-space is compact if and only if it is closed and bounded. (Heine–Borel theorem)
- A metric space (or uniform space) is compact if and only if it is complete and totally bounded.
- The product of any collection of compact spaces is compact. (Tychonoff's theorem, which is equivalent to the axiom of choice)
- A compact Hausdorff space is normal.
- Every continuous map from a compact space to a Hausdorff space is closed and proper. It follows that every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.
- A metric space (or more generally any first-countable uniform space) is compact if and only if every sequence in the space has a convergent subsequence.
- A topological space is compact if and only if every net on the space has a convergent subnet.
- A topological space is compact if and only if every filter on the space has a convergent refinement.
- A topological space is compact if and only if every ultrafilter on the space is convergent.
- A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.
- Every non-compact topological space X is a dense subspace of a compact space which has at most one point more than X. (Alexandroff one-point compactification)
- If the metric space X is compact and an open cover of X is given, then there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. (Lebesgue's number lemma)
- If a topological space has a sub-base such that every cover of the space by members of the sub-base has a finite subcover, then the space is compact. (Alexander's sub-base theorem)
- Two compact Hausdorff spaces X1 and X2 are homeomorphic if and only if their rings of continuous real-valued functions C(X1) and C(X2) are isomorphic. (Gelfand-Naimark theorem)

## Other forms of compactness

There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces. These include the following.- Sequentially compact: Every sequence has a convergent subsequence.
- Countably compact: Every countable open cover has a finite subcover. (Or, equivalently, every infinite subset has an ω-accumulation point.)
- Pseudocompact : Every real-valued continuous function on the space is bounded.
- Weakly countably compact (or limit point compact): Every infinite subset has an accumulation point.

While all these conditions are
equivalent for metric
spaces, in general we have the following
implications:

- Compact spaces are countably compact.
- Sequentially compact spaces are countably compact.
- Countably compact spaces are pseudocompact and weakly countably compact.

Not every countably compact
space is compact; an example is given by the first uncountable
ordinal with the order topology. Not every compact space is
sequentially compact; an example is given by 2[0,1], with the
product topology. A metric space is called pre-compact or totally
bounded if any sequence has a Cauchy subsequence; this can be
generalised to uniform
spaces. For complete metric spaces this is equivalent to
compactness. See relatively
compact for the topological version.

Another related notion which
(by most definitions) is strictly weaker than compactness is
local
compactness.

## Notes

## References

- Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology (1978) Springer-Verlag, New York

compact in Czech: Kompaktní
množina

compact in German: Kompakter
Raum

compact in Spanish: Espacio
compacto

compact in Esperanto:
Kompakta spaco

compact in Persian: مجموعه
فشرده

compact in French: Compacité
(mathématiques)

compact in Korean: 컴팩트
공간

compact in Italian: Spazio
compatto

compact in Hebrew:
קומפקטיות

compact in Dutch:
Compact

compact in Japanese:
コンパクト空間

compact in Polish: Przestrzeń
zwarta

compact in Portuguese: Espaço
compacto

compact in Russian:
Компактное пространство

compact in Swedish:
Kompakt

compact in Vietnamese:
Compact

compact in Turkish:
Tıkızlık

compact in Chinese:
紧集

compact in Classical Chinese:
緊集

# Synonyms, Antonyms and Related Words

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close-woven, clown white, coarct, coffer, coffin, coincident, cold cream,
combine, compact, compacted, compendious, comprehensive, compress, compressed, concentrate, concentrated, concise, concrete, concurrent, condense, condensed, cone, congest, congested, conjoint, conjugate, conjunct, consolidate, consolidated, consortium, constrict, constricted, constringe, contract, contracted, convention, cordial
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a deal, docked, draw, draw in, draw together,
drugstore complexion, duodecimo, dustproof, dusttight, elliptic, engage, entente, entente cordiale,
envelope, epigrammatic, etui, eye shadow, eyebrow pencil,
fast, file, file folder, filing box,
firm, folio, foundation, foundation cream,
full, gasproof, gastight, gluey, gnomic, greasepaint, hand cream,
hand lotion, handy,
hard, heavy, hermetic, hermetically sealed,
holster, hope chest,
housewife, hussy, hutch, impenetrable, impermeable, inclusive, instantaneous, integrate, jam, jam-packed, jammed, joined, joint, kit, knit, knitted, laconic, letter file, lightproof, lighttight, lip rouge,
lipstick, little, low, make a deal, makeup, mascara, massive, meaty, miniature, miniaturized, minikin, minimal, minuscule, monstrance, mudpack, mutual understanding,
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squeeze, squeezed, staunch, stipulate, stormproof, stormtight, strangle, strangled, strangulate, strangulated, subminiature, substantial, succinct, summary, swarming, synopsized, synoptic, taciturn, talcum, talcum powder, tea chest,
teeming, terse, thick, thick-growing, thickset, tight, till, tinderbox, to the point,
toy, transaction, transient, truncated, twelvemo, understanding, undertake, unite, vanishing cream, vanity
case, vasculum,
vest-pocket, viscid,
viscose, viscous, wallet, war paint, wasp-waisted,
water-repellant, waterproof, watertight, windproof, windtight, wrinkle, wrinkled