Compact space

A topological space each open covering of which contains a finite subcovering. The following statements are equivalent: 1) $ X $ is a non-empty compact space; 2) the intersection of any centred system of closed sets in $ X $ is non-empty; 3) the intersection of any maximal centred system of closed sets in $ X $ is non-empty; 4) the intersection of any decreasing totally ordered sequence of arbitrary cardinality of non-empty closed sets in $ X $ is non-empty; 5) each centred system of subsets of $ X $ has an accumulation point in $ X $; 6) each ultra-filter on $ X $ converges in $ X $; and 7) for each infinite subset $ M $ of $ X $ there exists a complete accumulation point in $ X $. A subspace of an $ n $- dimensional Euclidean space is compact if and only if it is closed and bounded. The concept of a compact topological space is fundamental in topology and modern functional analysis; certain fundamental properties of compact spaces (with numerous applications) are already considered in mathematical analysis, e.g. every real-valued continuous function defined on a compact space is bounded and attains its largest and its lowest value.

The term "compact space" is due to P.S. Aleksandrov, who made a very large contribution to the development of the theory of compact spaces. The fundamentals of the theory were laid by Aleksandrov and P.S. Urysohn in their Mémoire sur les espaces topologiques compacts.

The concept of a compact space was originally a strengthening of that of a compact space introduced by M. Fréchet: A non-empty topological space is compact in the original sense of the word (or countably compact, as they are now called) if it satisfies any one of the following equivalent statements: 1) each countable open covering of subsets of this space contains a finite subcovering; 2) the intersection of any countable centred system of non-empty subsets is non-empty; 3) the intersection of any countable decreasing sequence of non-empty closed sets is non-empty; 4) each countable centred system of subsets of the set of its subsets has an accumulation point; and 5) for each countable subset in it there exists a complete accumulation point.

However, subsequent development of mathematics and its applications showed that our concept of compactness (in Russian "bicompactness" ) is so much more important than the original concept of compactness that in its modern meaning the term is "compactness" , while compact spaces in the older sense of the word are denoted as countably compact. Both concepts are equivalent when applied to metric spaces.

A closed subspace of a compact space is a compact space. The topological product of any set of compact spaces is a compact space (Tikhonov's theorem, see Tikhonov product). A topological space that is the image of a compact space under a continuous mapping is a compact space. These properties of compact spaces show the stability of the class of compact spaces with respect to the operations which are fundamental to general topology, and applications of the concept of compactness are mostly based on such operations. Of special significance are compact spaces satisfying the Hausdorff separation axiom, they are called $ T _ {2} $- compacta. The topological product of any set of $ T _ {2} $- compacta is a $ T _ {2} $- compactum; a closed subspace of a $ T _ {2} $- compactum is a $ T _ {2} $- compactum; the image of a $ T _ {2} $- compactum under a continuous mapping into a Hausdorff space is a $ T _ {2} $- compactum. Below properties of $ T _ {2} $- compacta which are not displayed by arbitrary compact spaces are stated. Each $ T _ {2} $- compactum is normal and, a fortiori, a completely-regular space. The intersection of any countable set of open, everywhere-dense subsets in a $ T _ {2} $- compactum is everywhere-dense. An equivalent statement is: No non-empty $ T _ {2} $- compactum can be represented as the union of a countable set of nowhere-dense sets. $ T _ {2} $- compacta are characterized as those regular spaces that are closed in any Hausdorff space containing them. This is the key to a theorem according to which every continuous mapping of a compact space onto a Hausdorff space is closed. This theorem has an important consequence: Any one-to-one continuous mapping of a compact space onto a Hausdorff space is a homeomorphism.

Just like the class of $ T _ {2} $- compacta, so too the class of compact spaces is invariant with respect to transition to the space of closed subsets (taken with the Vietoris topology); moreover, the weight of the space does not increase. The space of non-empty closed subsets of the Cantor set $ C $ is homeomorphic to $ C $. A special role in the theory of compact spaces is played by the Tikhonov cubes $ I ^ { \tau } $ and by the generalized Cantor discontinua $ D ^ { \tau } $, defined, respectively, as topological products of the segment $ [ 0, 1] $ and as products of discrete two point spaces (or doublets), where $ \tau $ is an arbitrary cardinal number. If $ \tau = \aleph _ {0} $, one obtains the Hilbert cube. All $ I ^ { \tau } $ are $ T _ {2} $- compacta, and every $ T _ {2} $- compactum of weight not exceeding $ \tau $ is homeomorphic to some closed subspace of the cube $ I ^ { \tau } $. Thus, any $ T _ {2} $- compactum may be obtained from segments by only two operations: taking a topological product and transition to a closed subspace. The class of all subspaces of $ T _ {2} $- compacta can also be more exactly described as the class of all subspaces of the cubes $ I ^ { \tau } $. On the other hand, this is exactly the class of all completely-regular spaces.

The weight of a topological space — one of the most general topological invariants — becomes a particularly important characteristic in the case of $ T _ {2} $- compacta. A $ T _ {2} $- compactum is metrizable if and only if it has a countable base. At the same time, if the $ T _ {2} $- compactum $ Y $ is the image of a space $ X $ under a continuous mapping, the weight of $ Y $ is not larger than the weight of $ X $, i.e. there are no non-metrizable $ T _ {2} $- compacta among the continuous images of spaces with a countable base. The weight of a $ T _ {2} $- compactum $ Y = Y _ {1} \cup Y _ {2} $ is not larger than the larger of the weights of $ Y _ {1} $ and $ Y _ {2} $, i.e. no non-metrizable $ T _ {2} $- compactum can be represented as a sum of two spaces with a countable base. The two last-named facts are based on the concept of a net (of sets in a topological space) (or network). If a $ T _ {2} $- compactum has a net of cardinality $ \leq \tau $, it also has a base of cardinality $ \leq \tau $. In particular, any countable $ T _ {2} $- compactum has a countable base, is metrizable and is even homeomorphic to a closed subset of a segment. Metrizable $ T _ {2} $- compacta are often referred to as compacta. Any metric on a compactum is complete and totally bounded, and all metrizable spaces having this property are compacta (F. Hausdorff).

The topological product of a countable set of metrizable compacta and closed subsets of metrizable compacta are metrizable compacta. Each zero-dimensional metrizable compactum is homeomorphic to some compactum contained in the Cantor set (as a closed subset). Any metrizable compactum is a continuous image of the Cantor set (Aleksandrov). Each $ T _ {2} $- compactum is a continuous image of some zero-dimensional $ T _ {2} $- compactum.

Owing to the important role of the cubes $ I ^ { \tau } $ in topology, studies were carried out of the class of $ T _ {2} $- compacta which decompose into a product of compacta, as well as of the class of their continuous images. $ T _ {2} $- compacta that are continuous images of discontinua $ D ^ { \tau } $ are said to be dyadic. The class of dyadic $ T _ {2} $- compacta is quite extensive: All compacta, all cubes $ I ^ { \tau } $ and the spaces of all compact topological groups are dyadic $ T _ {2} $- compacta. The class of dyadic $ T _ {2} $- compacta is the smallest class of topological Hausdorff spaces that is closed with respect to topological products and continuous mappings, and which contains all $ T _ {2} $- compacta consisting of a finite number of points.

The simplest example of a non-dyadic $ T _ {2} $- compactum is an uncountable $ T _ {2} $- compactum with only one non-isolated point. Dyadic $ T _ {2} $- compacta have a number of remarkable properties; for instance, any disjoint system of non-empty open sets of a dyadic $ T _ {2} $- compactum is finite or countable (the Suslin property); any dyadic $ T _ {2} $- compactum satisfying the first axiom of countability is metrizable, i.e. has a countable base; the product of dyadic $ T _ {2} $- compacta is a dyadic $ T _ {2} $- compactum; a $ T _ {2} $- compactum which is a continuous image of a dyadic $ T _ {2} $- compactum is dyadic.

Ordered $ T _ {2} $- compacta, too, display specific properties. A connected separable ordered $ T _ {2} $- compactum has a countable base. An example of a connected separable (dyadic) $ T _ {2} $- compactum without a countable base is the cube $ I ^ { \tau } $ if $ \tau = \mathfrak c $( the cardinality of the continuum). Each $ T _ {2} $- compactum which is a continuous image of an ordered $ T _ {2} $- compactum has a base, the boundaries of the elements of which are compacta. The intersection of the class of dyadic $ T _ {2} $- compacta with the class of all $ T _ {2} $- compacta that are continuous images of ordered $ T _ {2} $- compacta consists of exactly all compacta.

Perfectly-normal $ T _ {2} $- compacta are important compact spaces. A normal space is said to be perfectly normal if each closed set in it is the intersection of countably many open sets. Every perfectly-normal $ T _ {2} $- compactum has the Suslin property. Any space with a countable net and which is a subspace of a perfectly-normal $ T _ {2} $- compactum has a countable base. The product of two perfectly-normal $ T _ {2} $- compacta need not be a perfectly-normal $ T _ {2} $- compactum: The space $ X \times X $ is a perfectly-normal $ T _ {2} $- compactum if and only if $ X $ is a compactum. However, the image of a perfectly-normal $ T _ {2} $- compactum and the product of a perfectly-normal $ T _ {2} $- compactum and a compactum are perfectly normal. A $ T _ {2} $- compactum is perfectly normal if and only if each one of its subspaces is finally compact. The problem of the existence of non-separable ordered perfectly-normal $ T _ {2} $- compacta is equivalent to the Suslin problem.

The study of connections between cardinal-valued topological invariants in the case of $ T _ {2} $- compacta forms the subject of numerous studies. If a base of a $ T _ {2} $- compactum is pointwise countable, it is countable. The cardinality of any uncountable $ T _ {2} $- compactum satisfying the first axiom of countability is equal to the cardinality of the continuum. If a $ T _ {2} $- compactum is sequential and satisfies the Suslin condition, its cardinality is not larger than that of the continuum. If a homogeneous $ T _ {2} $- compactum $ X $ is sequential, $ | X | \leq \mathfrak c $. A totally separable $ T _ {2} $- compactum of cardinality exceeding $ \mathfrak c $ has been constructed under a special assumption.

There exists a number of theorems about $ T _ {2} $- compacta related to the concept of universality (cf. Universal space). For each cardinal number $ \tau $ there exists a $ T _ {2} $- compactum of weight $ \tau $ such that each $ T _ {2} $- compactum of a weight not exceeding $ \tau $ is imbedded in it by a homeomorphism. Thus, $ I ^ { \tau } $ displays this property. A similar result is obtained in the case in which the weight and the dimension dim or Ind are fixed: Whatever the cardinal number $ \tau $ and the natural number $ n $, there exist $ T _ {2} $- compacta $ \Pi ^ {n \tau } $ and $ \Psi ^ {n \tau } $ of weight $ \tau $ and dimension $ n $( $ \mathop{\rm dim} \Pi ^ {n \tau } $ and $ \mathop{\rm Ind} \Psi ^ {n \tau } $) such that any $ T _ {2} $- compactum of weight not exceeding $ \tau $ and dimension not exceeding $ n $ can be imbedded in $ \Pi ^ {n \tau } $( or in $ \Psi ^ {n \tau } $) by a homeomorphism. $ D ^ { \tau } $ may be used as $ \Pi ^ {\omicron \tau } $ and $ \Psi ^ {\omicron \tau } $. If it is assumed that the generalized continuum hypothesis is valid, a dual result is obtained: For each cardinal number $ \tau $ there exists a zero-dimensional $ T _ {2} $- compactum $ X $ of weight $ \tau $ that can be continuously mapped onto each $ T _ {2} $- compactum whose weight does not exceed $ \tau $.

The part of general topology dealing with continuous mappings of $ T _ {2} $- compacta has been especially intensively studied and numerous results are available. Thus, if $ X $ and $ Y $ are $ T _ {2} $- compacta and if $ \phi : X \rightarrow Y $ is a continuous mapping with $ \phi ( X) = Y $, then there exists a subspace $ X _ {1} $ closed in $ X $ for which $ \phi ( X _ {1} ) = Y $ and such that the restriction $ \phi \mid _ {X _ {1} } : X _ {1} \rightarrow Y $ is an irreducible mapping. This explains the fundamental role played by irreducible mappings. For any given $ T _ {2} $- compactum $ X $ there exists in the set $ N( X) $ of all $ T _ {2} $- compacta $ Z $ which admit an irreducible mapping onto $ X $ a $ T _ {2} $- compactum $ \dot{X} $ that can be irreducibly mapped onto any $ T _ {2} $- compactum in $ N( X) $. This $ T _ {2} $- compactum $ \dot{X} $ is said to be the absolute of the $ T _ {2} $- compactum $ X $( cf. Absolute of a regular topological space); it is defined uniquely up to a homeomorphism. A $ T _ {2} $- compactum is homeomorphic to its absolute if and only if it is extremally disconnected (cf. Extremally-disconnected space). There are a lot of disconnected $ T _ {2} $- compacta; to each $ T _ {2} $- compactum corresponds its absolute, which is an extremally-disconnected $ T _ {2} $- compactum. The irreducible mappings of $ T _ {2} $- compacta are defined by the homeomorphisms of their absolutes. They have a very characteristic structure; in particular, all extremally-disconnected $ T _ {2} $- compacta are non-homogeneous. Two $ T _ {2} $- compacta are said to be co-absolute if their absolutes are homeomorphic. Here, one of the main problems is to find effective intrinsic criteria of co-absoluteness of topological spaces. It should be noted that the natural domain of application of the theory of absolutes is much more extensive than the class of $ T _ {2} $- compacta.

Irreducible mappings are also important in contexts other than the concept of the absolute. It is known (A.N. Kolmogorov) that all irreducible mappings of one metrizable compactum into another contain an everywhere-dense set of one-to-one points. If a dyadic $ T _ {2} $- compactum $ X $ is irreducibly mapped onto a $ T _ {2} $- compactum $ Y $ of weight $ \tau $, the weight of $ X $ is equal to $ \tau $.

Open mappings of $ T _ {2} $- compacta are also important. If two infinite $ T _ {2} $- compacta are related by an open finite-to-one mapping, their weights are equal. However, there exists an open, countably-to-one mapping of a non-metrizable, perfectly-normal $ T _ {2} $- compactum onto a metrizable compactum. For any such mapping there is an everywhere-dense set of points certain neighbourhoods of which are mapped homeomorphically.

$ T _ {2} $- compacta were extensively studied in the framework of dimension theory. The relation $ \mathop{\rm dim} X \leq \mathop{\rm ind} X $ is true for any $ T _ {2} $- compactum $ X $( Aleksandrov). There exists a $ T _ {2} $- compactum $ X $ satisfying the first axiom of countability (and, consequently, with cardinality not exceeding that of the continuum) for which $ { \mathop{\rm dim} } X \neq { \mathop{\rm ind} } X \neq { \mathop{\rm Ind} } X $, but for all perfectly-normal $ T _ {2} $- compacta $ { \mathop{\rm ind} } X = { \mathop{\rm Ind} } X $. It is of interest to contrast the following two theorems: 1) open countably-to-one mappings do not increase the dimension of $ T _ {2} $- compacta; 2) the representation theorem, viz. each $ T _ {2} $- compactum of positive dimension is the image of some one-dimensional $ T _ {2} $- compactum under an open-closed continuous mapping in which the inverse image of any point is zero-dimensional. The following factorization theorem is important: Let $ f: X \rightarrow Y $ be a continuous mapping, let $ X $ and $ Y $ be $ T _ {2} $- compacta, $ f( X) = Y $, let $ { \mathop{\rm dim} } X \leq n $, and let the weight of $ Y $ be at most $ \tau $; then there exist a $ T _ {2} $- compactum $ Z $ and continuous mappings $ \phi : X \rightarrow Z $ and $ \psi : Z \rightarrow Y $ for which $ f = \psi \phi $, $ { \mathop{\rm dim} } Z \leq n $, and the weight of $ Z $ is less than or equal to $ \tau $.

The relations between compact spaces and arbitrary topological spaces form the subject of the theory of compactifications (cf. Compactification).

References

[1]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) MR0168706
[2]	P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian) MR0365524 Zbl 0441.55002
[3]	R. Engelking, "Outline of general topology" , North-Holland (1968) (Translated from Polish) MR0230273 Zbl 0157.53001
[4]	P.S. Aleksandrov, P.S. Uryson, "On compact topological spaces" Trudy Mat. Inst. Steklov. , 31 (1950) (In Russian) MR0043445 Zbl 0228.54017
[5]	P.S. Aleksandrov, "On bicompact extensions of topological spaces" Mat. Sb. , 5 (47) : 2 (1939) pp. 403–424 (In Russian) (German abstract) MR0097787
[6]	A.V. Arkhangel'skii, "An addition theorem for weights of sets lying in bicompacta" Dokl. Akad. Nauk SSSR , 126 : 2 (1959) pp. 239–241 (In Russian)
[7]	A.V. Arkhangel'skii, "Bicompact sets and the topology of spaces" Trans. Moscow Math. Soc. , 13 (1965) pp. 1–62 Trudy Moskov. Mat. Obshch. , 13 (1965) pp. 3–55 Zbl 0162.26602
[8]	A.V. Arkhangel'skii, "Approximation of the theory of dyadic bicompacta" Soviet Math. Dokl. , 10 : 4 (1969) pp. 151–154 Dokl. Akad. Nauk SSSR , 184 : 4 (1969) pp. 767–771 Zbl 0182.25502
[9]	A.S. Mishchenko, "Spaces with point countable base" Soviet Math. Dokl. , 3 : 3 (1962) pp. 855–858 Dokl. Akad. Nauk SSSR , 144 : 5 (1962) pp. 985–988 Zbl 0122.17304
[10]	A.M. Gleason, "Projective topological spaces" Illinois J. Math. , 2 : 4A (1958) pp. 482–489 MR0121775 Zbl 0083.17401
[11]	A.V. Arkhangel'skii, "On the cardinality of bicompacta satisfying the first axiom of countability" Soviet Math. Dokl. , 10 : 4 (1969) pp. 951–955 Dokl. Akad. Nauk SSSR , 187 : 5 (1969) pp. 967–970 Zbl 0191.20903
[12]	S. Mardešić, "Images of ordered compacta and locally peripherically metric spaces" Pacific J. Math , 23 (1967) pp. 557–568
[13]	A.N. Kolmogorov, Dokl. Akad. Nauk SSSR , 30 : 6 (1941) pp. 477–479 MR2395530
[14]	A.V. Arkhangel'skii, "Factorization of mappings according to weight and dimension" Soviet Math. Dokl. , 8 : 3 (1967) pp. 731–734 Dokl. Akad. Nauk SSSR , 174 : 6 (1967) pp. 1243–1246 Zbl 0174.54202
[15]	W. Hanf, "On some fundamental problems concerning isomorphism of Boolean algebras" Math. Scand. , 5 (1957) pp. 205–217 MR0108451 Zbl 0081.26101
[16]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) MR785749 Zbl 0568.54001

Comments

The terminology on compactness can be confusing. The Russian usage can be seen from the text. In particular, the word "compact" often means "countably compact" in the Russian literature.

In the West "compact" is used for both compact and $ T _ {2} $- compact, and the former is sometimes called quasi-compact. In topology the majority of Western authors equate compact and compact Hausdorff ( $ T _ {2} $- compact), because the latter spaces are much better behaved; on the other hand in, e.g., algebraic geometry the term compact does not as a rule include $ T _ {2} $.

The defining property for compactness, i.e. every open covering has a finite subcovering, is also often referred to as the Heine–Borel property, cf. also Borel–Lebesgue covering theorem.

The property "finally compact" is usually called Lindelöf (or Lindelöf compact) in the West (cf. Lindelöf space).

A family of sets $ {\mathcal F} $ is centred if the intersection of any finite number of elements of $ {\mathcal F} $ is non-empty; $ {\mathcal F} $ is then also said to have the finite intersection property. The words directed and filtered also occur in this connection. More precisely and more generally, let $ ( A , < ) $ be a partially ordered set. The set is said to be upper directed if for all $ a , b \in A $ there is a $ c \in A $ such that $ a < c $, $ b < c $; it is lower directed if for all $ a , b \in A $ there is a $ c \in A $ such that $ c < a $, $ c < b $. The phrases filtered to the right and filtered to the left also occur to describe these two notions. The set $ ( A , < ) $ is directed if it is both upper and lower directed. The word directed is also used to denote just upper directed (depending on the context); the same usage also applies to "filtered" .

There are a lot of notions in topology strongly related to compactness [16]. The index of compactness of a topological space $ X $ is the least cardinal number $ \tau $ such that each open covering of $ X $ has a subcovering of cardinality $ \leq \tau $. Thus, a space is Lindelöf compact if its index of compactness is $ \aleph _ {0} $. For the notions sequential compactness, local compactness and pseudo-compactness see, respectively, Sequentially-compact space; Locally compact space; Pseudo-compact space. For the notion paracompactness and various related concepts such as weak and strong paracompactness, hereditarily paracompactness, star-paracompactness, $ \tau $- paracompactness, countable paracompactness, see Paracompact space. A space is $ \sigma $- compact if it is a countable union of compact subspaces. If $ \tau $ is a cardinal number, a space is called $ \tau $- compact if every covering of cardinality at most $ \tau $ has a finite subcovering. This generalizes countable compactness of course. For the notions of $ [ a , b ] $- compactness, initial compactness and final compactness see (also) Compactness, countable. A basis $ \{ U _ \alpha \} _ {\alpha \in A } $ of a space $ X $ is $ \pi $- compact if for all $ U _ \alpha $ the boundary or frontier $ \partial U _ \alpha = \textrm{ Fr } U _ \alpha = \overline{U}\; _ \alpha \cap ( \overline{ {X \setminus U _ \alpha }}\; ) $ is compact. A space is called $ \pi $- compact if it has a $ \pi $- compact basis. This is also called peripherical compactness. Finally, for the notion of a real compact space cf. Hewitt realcompactification.

The Mémoire of Aleksandrov and Urysohn referred to in the main article above is [a3].

References

[a1]	R. Engelking, "General topology" , PWN (1977) (Translated from Polish) (Revised and extended version of [3] above) MR0500779 MR0500780 Zbl 0373.54002
[a2]	S. Mardešić, "On covering dimension and inverse limits of compact spaces" Illinois J. Math. , 4 (1960) pp. 278–291 MR0116306 Zbl 0094.16902
[a3]	P.S. Aleksandrov, P. Urysohn, "Mémoire sur les espaces topologiques compacts" , Koninkl. Nederl. Akad. Wetensch. , Amsterdam (1929)