出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/11/09 09:54:05」(JST)
In the mathematical discipline of general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, contained within some finite cube of the appropriate dimension). This notion is generalized to more general topological spaces in various ways. For instance, a space is sequentially compact if any infinite sequence of points sampled from the space must eventually, infinitely often, get arbitrarily close to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sense if and only if it is closed and bounded. Examples include a closed interval or a rectangle. Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … get arbitrarily close to 0. (Also, some get arbitrarily close to 1.) The same set of points would not have, as a limit point, any point of the open unit interval; so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points 0, 1, 2, 3, … has no sub-sequence that ultimately gets arbitrarily close to any given real number.
Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions. The term compact was introduced into mathematics by Maurice Fréchet in 1906 as a distillation of this concept. Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation. A typical application is furnished by the Arzelà–Ascoli theorem and in particular the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.
Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces. In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that "cover" the space in the sense that each point of the space must lie in some set contained in the family. The standard unqualified use of the term compact in mathematics usually means compactness in this latter sense. This more subtle definition exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this 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.
An example of a compact space is the unit interval [0,1] of real numbers. If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point in that interval. For instance, the odd-numbered terms of the sequence 1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, … get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1. 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. It is also crucial that the interval be bounded, since in the interval [0,∞) one could choose the sequence of points 0, 1, 2, 3, …, of which no sub-sequence ultimately gets arbitrarily close to any given real number.
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. 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. 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.
Compactness generalizes many important properties of closed and bounded intervals in the real line; that is, intervals of the form [a, b] for real numbers a and b. For instance, any continuous function defined on a compact space into an ordered set (with the order topology) such as the real line is bounded. 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.
Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set. This puts a fine point on the idea of taking "steps" in a space. Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.
In general topological spaces, however, the different notions of compactness are not equivalent, and the most useful notion of compactness—originally called bicompactness—involves families of open sets that cover the space in the sense that each point of the space must lie in some set contained in the family. 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. That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take information that is known locally—in a neighbourhood of each point of the space—and to extend it to information that holds globally throughout the space. An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous: here continuity is a local property of the function, and uniform continuity the corresponding global property.
Formally, a topological space X is called compact if each of its open covers has a finite subcover. Otherwise it is called non-compact. Explicitly, this means that for every arbitrary collection
of open subsets of X such that
there is a finite subset J of A such that
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. A compact set is sometimes referred to as a compactum, plural compacta.
A space X is compact if its natural extension *X (for example, an ultrapower) has the property that every point of *X is infinitely close to a suitable point of . For example, an open real interval X=(0,1) is not compact because its hyperreal extension *(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point of X.
A subset K of a topological space X is called compact if it is compact in the induced topology. Explicitly, this means that for every arbitrary collection
of open subsets of X such that
there is a finite subset J of A such that
In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano (1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a limit point. Bolzano's proof relied on the method of bisection: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts until it closes down on the desired limit point. The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.^{[1]}
In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points. The idea of regarding functions as themselves points of a generalized space dates back to the investigations of Giulio Ascoli and Cesare Arzelà.^{[2]} The culmination of their investigations, the Arzelà–Ascoli theorem, was a generalization of the Bolzano–Weierstrass theorem to families of continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergent sequence of functions from a suitable family of functions. The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point". Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and Erhard Schmidt. For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense of mean convergence—or convergence in what would later be dubbed a Hilbert space. This ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property and coined the term compactness to refer to this general phenomenon.
However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis. In 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. The significance of this lemma was recognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (1904). The Heine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.
This property was significant because it allowed for the passage from local information about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function). This sentiment was expressed by Lebesgue (1904), who also exploited it in the development of the integral now bearing his name. Ultimately the Russian school of point-set topology, under the direction of Pavel Alexandrov and Pavel Urysohn, formulated Heine–Borel compactness in a way that could be applied to the modern notion of a topological space. Alexandrov & Urysohn (1929) showed that the earlier version of compactness due to Fréchet, now called (relative) sequential compactness, under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers. It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.
Some theorems related to compactness (see the glossary of topology for the definitions):
Assuming the axiom of choice, the following are equivalent.
For any subset A of Euclidean space R^{n}, the following are equivalent:
In practice, the condition (5) is easiest to verify, for example a closed interval or closed n-ball. Note that, in a metric space, every compact subset is closed and bounded. However, the converse may fail in non-Euclidean R^{n}. 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.
Let X be a topological space and C(X) the ring of real continuous functions on X. For each p∈X, the evaluation map
given by ev_{p}(f)=f(p) is a ring homomorphism. The kernel of ev_{p} is a maximal ideal, since the residue field C(X)/ker ev_{p} is the field of real numbers, by the first isomorphism theorem. A topological space X is pseudocompact if and only if every maximal ideal in C(X) has residue field the real numbers. For completely regular spaces, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.^{[11]} There are pseudocompact spaces that are not compact, though.
In general, for non-pseudocompact spaces there are always maximal ideals m in C(X) such that the residue field C(X)/m is a (non-archimedean) hyperreal field. The framework of non-standard analysis allows for the following alternative characterization of compactness:^{[12]} a topological space X is compact if and only if every point x of the natural extension *X is infinitely close to a point x_{0} of X (more precisely, x is contained in the monad of x_{0}).
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.
While all these conditions are equivalent for metric spaces, in general we have the following implications:
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 (Scarborough & Stone 1966, Example 5.3).
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.
Generalizations of compactness include H-closed and the property of being an H-set in a parent space. A Hausdorff space is H-closed if every open cover has a finite subfamily whose union is dense. Whereas we say X is an H-set of Z if every cover of X with open sets of Z has a finite subfamily whose Z closure contains X.
This article incorporates material from Examples of compact spaces on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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