Compactification

compact extension

An extension of a topological space which is a compact space. A compactification exists for any topological space, and any $ T _ {1} $- space has a compactification which is a $ T _ {1} $- space, but Hausdorff compactifications of completely-regular spaces (cf. Completely-regular space) are of the greatest interest. A compactification usually means a Hausdorff compactification, but arbitrary compactifications may also be considered. It was proved by P.S. Aleksandrov [1] that all locally compact Hausdorff spaces may be completed to a $ T _ {2} $- compactum by the addition of one point (cf. Aleksandrov compactification). P.S. Urysohn [2] proved that every normal space with a countable base can be imbedded in the Hilbert cube, which implies that it has a compactification of countable weight [2]. The term "compactification" was first introduced by A.N. Tikhonov [3], who defined the class of completely-regular spaces and proved that completely-regular spaces and only such spaces have a Hausdorff compactification, a completely-regular space of weight $ \tau $ having a Hausdorff compactification of weight $ \tau $.

Two compactifications $ b _ {1} X $ and $ b _ {2} X $ of a space $ X $ are said to be equivalent ( $ b _ {1} X \simeq b _ {2} X $) if there exists a homeomorphism $ f: b _ {1} X \rightarrow b _ {2} X $ which is the identity on $ X $. Often it is the imbedding $ i: X \rightarrow bX $ itself which is called a compactification. If this definition is accepted, two extensions $ i _ {1} : X \rightarrow b _ {1} X $ and $ i _ {2} : X \rightarrow b _ {2} X $ are equivalent if there exists a homeomorphism $ f: b _ {1} X \rightarrow b _ {2} X $ such that $ f \circ i _ {1} = i _ {2} $. Equivalent compactifications are usually not distinguished, and a class of mutually equivalent compactifications of a space $ X $ is viewed as a compactification of that space. In such a case one can speak of the set $ B( X) $ of Hausdorff compactifications of a given (completely-regular) space $ X $, since the cardinality of any Hausdorff extension of $ X $ is at most $ 2 ^ {2 ^ {| X | } } $, while the topologies on a given set $ Y $ also form a set of cardinality $ \leq 2 ^ {2 ^ {| Y | } } $.

A compactification $ b _ {2} X $ follows a compactification $ b _ {1} X $( $ b _ {1} X \leq b _ {2} X $) if there exists a continuous mapping $ f: b _ {2} X \rightarrow b _ {1} X $ which is the identity on $ X $. The successor relation converts $ B( X) $ into a partially ordered set. E. Čech [4] and M.H. Stone [5] showed that the set $ B( X) $ contains a largest element $ \beta X $, the Stone–Čech compactification (or maximal compactification).

The problem of the intrinsic description of all Hausdorff compactifications of a given completely-regular space $ X $ has been solved [6] by constructing the compactifications of an arbitrary proximity space, thus proving that to each proximity $ \delta $ on $ X $ which is compatible with the topology there corresponds a unique compactification $ b _ \delta X $ which induces the initial proximity $ \delta $ on $ X $, i.e.

$$ A {\overline \delta \; } B \iff \ [ A] _ {b _ \delta X } \cap [ B] _ {b _ \delta X } = \emptyset . $$

The maximal compactification $ \beta X $ is generated by the following proximity $ \delta $:

$$ A {\overline \delta \; } B \iff A \textrm{ and } B \textrm{ are functionally separable }. $$

The Aleksandrov compactification $ \alpha X $ of a locally compact Hausdorff space $ X $ is generated by the proximity $ \delta $:

$$ A {\overline \delta \; } B \iff A \textrm{ and } B \ \textrm{ have non\AAh intersecting } \ $$

$$ \textrm{ closures, at least one of which is compact }. $$

The correspondence $ \delta \rightarrow b _ \delta $ is an isomorphism between the partially ordered set of proximities on $ X $ which are compatible with the topology, and the set $ B( X) $. The correspondence $ \delta \rightarrow b _ \delta $ is extended to a functor from the category of spaces with a proximity that is compatible with the topology, with proximally continuous mappings, into the category of $ T _ {2} $- compacta, with continuous mappings.

A major part of the theory of compactifications is concerned with methods of constructing them. It was shown by Tikhonov that on each completely-regular space $ X $ of weight $ \tau $ there exists a set of functions $ f _ \alpha : X \rightarrow I _ \alpha $ of cardinality $ \tau $ such that their diagonal product realizes an imbedding of $ X $ into the cube $ I ^ \tau = \prod _ \alpha I _ \alpha $( cf. Tikhonov cube). After this, a compactification in $ X $ of weight $ \tau $ is obtained as the closure of $ fX $ in $ I ^ \tau $. Čech constructed the maximal compactification of a space $ X $ using the diagonal product of all continuous functions $ f _ \alpha : X \rightarrow [ 0, 1] $. Stone constructed the maximal compactification by using Boolean algebras and rings of continuous functions.

One of the fundamental methods in compactification theory is Aleksandrov's method of centred systems of open sets [7], which was initially used for the construction of the maximal compactification, and was subsequently extensively utilized by many mathematicians. Thus, it was found that any Hausdorff extension of an arbitrary Hausdorff space $ X $ can be realized as a space of centred systems of sets open in $ X $. The method of centred systems was utilized to construct an isomorphism between the set of proximities on a completely-regular space and the set of all its Hausdorff compactifications. The method was applied to the construction of Hausdorff compactifications of $ X $ from a subordination given on it.

H. Wallman [9] constructed the maximal compactification of a normal space $ X $ as the space of maximal centred systems of closed sets of this space. The space $ \omega X $ of maximal centred systems of closed sets of a $ T _ {1} $- space $ X $ is its $ T _ {1} $- compactification and is called the Wallman compactification. This compactification, like the Stone–Čech compactification, differs from other compactifications by the similarity between the combinatorial construction and the extendable space, the maximality (in a certain sense), and the possibility of extending continuous mappings.

The method of centred systems of closed sets makes it possible to generalize the Wallman compactification. In a completely-regular space $ X $ let there be given a base of closed sets $ \mathfrak B $ which is a ring of sets, i.e. contains the intersection and the union of any two elements in it. The base $ \mathfrak B $ is said to be normal if: 1) for any point $ x \in X $ and any element $ B \in \mathfrak B $ not containing this point there exist elements $ B _ {1} $ and $ B _ {2} $ of the base such that $ B _ {1} \cup B _ {2} = X $, $ x \in X \setminus B _ {1} $, $ B \subset X \setminus B _ {2} $; and 2) for any two elements $ B _ {1} , B _ {2} \in \mathfrak B $ there exist elements $ B _ {1} ^ { \prime } , B _ {2} ^ { \prime } \in \mathfrak B $ such that $ X = B _ {1} ^ { \prime } \cup B _ {2} ^ { \prime } $, $ B _ {1} \subset X \setminus B _ {1} ^ { \prime } $, $ B _ {2} \subset X \setminus B _ {2} ^ { \prime } $. The space of maximal centred systems of a normal base-ring with the standard given base of closed sets on it is a Hausdorff compactification of $ X $, known as a compactification of Wallman type; all Hausdorff compactifications are of Wallman type (Ul'yanov's theorem, cf. [22]).

Other methods of constructing compactifications include: the method of maximal ideals of the rings of continuous functions [11]; the method of completion of pre-compact uniform structures (cf. [12] and Completion of a uniform space); and the method of projective spectra (cf. Projective spectrum of a ring) [10]. It has been shown in this connection that the least upper bound of the maximal finite spectrum of any $ T _ {1} $- space $ X $ is its Wallman compactification $ \omega X $, and this bound coincides with the maximal compactification $ \beta X $ if and only if $ X $ is a quasi-normal space.

The importance of the theory of compactifications is explained by the fundamental role of compact spaces in topology and functional analysis. The possibility of imbedding a topological space in a $ T _ {2} $- compactum makes it possible to describe many properties of completely-regular spaces in terms of properties of $ T _ {2} $- compacta, which are usually simpler. Thus, normal spaces satisfying the first axiom of countability are homeomorphic if and only if their maximal compactifications are homeomorphic. Hence, the study of normal spaces satisfying the first axiom of countability can be reduced, in principle, to the study of $ T _ {2} $- compacta. The topological invariants of an extendable space can very often be expressed in a simple manner in terms of imbeddings of the space in its compactifications (cf. Feathered space; Completeness (in topology); Normally-imbedded subspace). Thus, for a space $ X $ to be a space of countable type, i.e. a space in which any $ T _ {2} $- compactum is contained in a $ T _ {2} $- compactum of countable character, it is necessary and sufficient for some (and hence, for all) compactifications $ bX $ that the remainder $ bX \setminus X $ be finally compact. The spaces $ X $ of countable type are also interesting because they are normally adjacent to the remainder in all their compactifications $ bX $, which means that any two non-intersecting sets, closed in the remainder, have neighbourhoods which do not intersect in $ bX $. As regards imbedding in compactifications, it is finally-compact spaces which are dual with spaces of countable type. A space $ X $ is finally compact if and only if one (and hence all) of its compactifications $ bX $ have the following property: For any $ T _ {2} $- compactum $ \Phi \subset bX \setminus X $ there exists in the remainder a $ T _ {2} $- compactum $ F $ containing it, which has countable character in $ bX $.

Compactifications are particularly important in dimension theory. This is explained, in particular, by the equations

$$ \mathop{\rm dim} \beta X = \mathop{\rm dim} X,\ \ \mathop{\rm Ind} \beta X = \ \mathop{\rm Ind} X, $$

which are valid for every normal space, and by the equation

$$ \mathop{\rm ind} \beta X = \ \mathop{\rm ind} X $$

for a perfectly-normal space $ X $. One of the first theorems regarding the dimensional properties of compactifications was the theorem according to which any $ n $- dimensional normal space with a countable base has a Hausdorff compactification of the same (countable) weight and the same dimension [16]. It has been proved [20] that only the peripherally-compact spaces (cf. Peripherically-compact space) among the normal spaces $ X $ with a countable base have a compactification $ bX $ with zero-dimensional (in the sense of the dimension ind) remainder $ bX \setminus X $( cf. Freudenthal compactification). Under such compactifications of this space there is a largest. These two results were the starting point of a large number of studies. Thus, it was proved [8] that for any completely-regular space $ X $ of weight $ \tau $ with $ \mathop{\rm dim} \beta X \leq n $, in particular for any normal space $ X $ of weight $ \tau $ with $ \mathop{\rm dim} X \leq n $, there exists a compactification $ bX $ of weight $ \tau $ and dimension $ \mathop{\rm dim} bX \leq n $. On the other hand, a completely-regular space $ X $ is peripherally compact if and only if $ X $ has a compactification with its remainder zero-dimensionally imbedded in it [8]. The remainder $ bX \setminus X $ is said to be zero-dimensionally imbedded in $ bX $( or, relatively zero-dimensional in $ bX $) whenever there exists a base $ \mathfrak B $ of $ bX $ such that

$$ ( bX \setminus X) \cap \mathop{\rm fr} _ {bX} \Gamma = \emptyset $$

for all $ \Gamma \in \mathfrak B $, where $ \mathop{\rm fr} _ {bX} \Gamma = \mathop{\rm cl} _ {bX} ( \Gamma ) \setminus \Gamma $, the frontier (or boundary) of $ \Gamma $.

Of obvious importance in the theory of compactifications are perfect compactifications (cf. Perfect compactification) . All perfect compactifications $ bX $ of a space $ X $ are monotone images of the maximal compactification $ \beta X $( in particular, $ \beta X $ itself is perfect as well) and have, like $ \beta X $, a combinatorial structure similar to $ X $, but, unlike in the case of $ \beta X $, $ \mathop{\rm dim} X = \mathop{\rm dim} bX $ is not always valid, not even for metric spaces $ X $. Whereas $ \beta X $ is the largest perfect compactification, a minimal perfect compactification exists only if $ X $ has a compactification with a punctiform remainder (in particular, if $ X $ is peripherally compact). The minimal perfect compactification is unique in having a punctiform remainder, and is the largest of all extensions with a punctiform remainder.

The concept of a compactification is useful in the study of the dimension of the remainder. If a metric space $ X $ with a countable base is imbedded in a compactum $ bX $ with remainder $ bX \setminus X $ of dimension $ \leq n $, there exists in $ X $ an (open) base such that the intersection of the boundaries of any of its $ n + 1 $ elements is compact . This condition is not sufficient for the space $ X $ to have a compactification $ bX $ with $ \mathop{\rm dim} bX = \mathop{\rm dim} X $ and $ \mathop{\rm dim} ( bX \setminus X) \leq n $. Moreover, if $ bX $ is a perfect compactification of $ X $, if $ \mathop{\rm dim} bX = n $, and if $ \mathop{\rm dim} \Phi \leq n- 1 $ for any compact set $ \Phi \subset bX \setminus X $, then $ \mathop{\rm dim} vX \geq n $ for any compactification $ vX $ with a punctiform remainder . Both perfect and maximal compactifications are of interest in the context of possible extensions of mappings. Thus, in particular, if the spaces $ X $ and $ Y $ have minimal perfect extensions $ \mu X $ and $ \mu Y $, any perfect mapping $ f: X \rightarrow Y $ can be extended to yield a mapping $ \overline{f}\; : \mu X \rightarrow \mu Y $.

A topological space $ X $ of weight $ \tau $ is zero-dimensional (i.e. $ \mathop{\rm ind} X = 0 $) if and only if [16] it has a zero-dimensional compactification $ bX $ of weight $ \tau $, so that a space $ X $ of weight $ \tau $ with $ \mathop{\rm Ind} X = 0 $ has a compactification of equal weight and equal dimension. In the case of a completely-regular space $ X $ with $ \mathop{\rm Ind} \beta X \leq n $ there exists a compactification $ bX $ with $ wbX = wX $ and $ \mathop{\rm Ind} bX \leq n $, and this statement is valid for transfinite values of $ \mathop{\rm Ind} \beta X $ as well ( $ wY $ denotes the weight of $ Y $). It follows that a strongly-paracompact metric space $ X $ has a compactification $ bX $ such that $ wbX = wX $, $ \mathop{\rm dim} bX = \mathop{\rm ind} bX = \mathop{\rm dim} X $, and there exists a space $ X $ such that $ \mathop{\rm ind} bX > \mathop{\rm ind} X $ for all its compactifications in $ X $[21].

There are several theorems concerning compactifications of infinite-dimensional spaces. Thus, the maximal compactification $ \beta X $ of a normal $ S $- weakly infinite-dimensional space $ X $ is weakly infinite-dimensional [16]. Any completely-regular space $ X $ of weight $ \tau $ with a weakly infinite-dimensional compactification (in particular, any normal $ S $- weakly infinite-dimensional space $ X $ of weight $ \leq \tau $) has a weakly infinite-dimensional compactification of weight $ \leq \tau $. In these theorems it is not allowed to replace the $ S $- weak infinite dimension by the $ A $- weak infinite dimension (cf. Weakly infinite-dimensional space). Thus, all compactifications of increasing sums of cubes $ Q ^ \omega $( subsets of the Hilbert cube $ Q ^ \infty $, consisting of points having only a finite number of non-zero coordinates) are strongly infinite-dimensional spaces .

Yu.M. Smirnov

studied the problems connected with the dimension dim of the remainders of compactifications of proximity spaces and completely-regular spaces. If a proximity space $ P $ is normally adjacent to $ cP \setminus P $, where $ cP $ is the (unique) compactification of $ P $, then $ \mathop{\rm dim} ( cP \setminus P) $ is equal to the smallest of the numbers $ k $ such that a bordering of multiplicity $ \leq k + 1 $ can be inscribed into each extended bordering (cf. Bordering of a space). A space $ X $ of countable type has a compactification $ bX $ with a remainder of dimension $ \leq n $ if and only if a bordering structure of multiplicity $ \leq n + 1 $, with the basis property, exists in $ X $. Moreover, a consequence of the existence of a compactification with a remainder of dimension $ \leq n $ in a given space $ X $ is the existence of a compactification $ bX $ of weight $ wbX = wX $ with a remainder of dimension $ \leq n $.

The partially ordered set $ B( X) $ of all Hausdorff compactifications of a space $ X $ is a complete semi-lattice (with respect to the operation of taking the supremum). The set $ B( X) $ is a complete lattice if and only if $ X $ is a locally compact space. If the spaces $ X $ and $ Y $ are locally compact, the lattices $ B( X) $ and $ B( Y) $ are isomorphic if and only if the remainders $ \beta X \setminus X $ and $ \beta Y \setminus Y $ are homeomorphic [18]. Conditions to be met by a (perfect) mapping $ f: X \rightarrow Y $ for the lattices $ B( X) $ and $ B( Y) $ to be isomorphic, are unknown. The compactifications of perfect irreducible inverse images of a space $ X $ are described by $ \theta $- proximities (cf. Proximity) on the space $ X $ and form a complete semi-lattice with respect to a naturally definable order [19]. Compactifications of perfect irreducible inverse images of a space $ X $ are also connected with $ H $- closed extensions of $ X $.

References

[1]	P.S. Aleksandrov, P. Urysohn, "Mémoire sur les espaces topologiques compacts" , Koninkl. Nederl. Akad. Wetensch. , Amsterdam (1929)
[2]	P.S. Urysohn, "Works on topology and other areas of mathematics" , 1–2 , Moscow-Leningrad (1951) (In Russian)
[3]	A.N. [A.N. Tikhonov] Tikhonoff, "Ueber die topologische Erweiterung von Räumen" Math. Ann. , 102 (1929) pp. 544–561
[4]	E. Čech, "On bicompact spaces" Ann. of Math. (2) , 38 : 4 (1937) pp. 823–844
[5]	M.H. Stone, "Applications of the theory of Boolean rings to general topology" Trans. Amer. Math. Soc. , 41 (1937) pp. 375–481
[6]	Yu.M. Smirnov, "On proximity spaces" Mat. Sb. , 31 (73) : 3 (1952) pp. 543–574 (In Russian)
[7]	P.S. Aleksandrov, "On bicompact extensions of topological spaces" Mat. Sb. , 5 (47) : 2 (1939) pp. 403–424 (In Russian) (German abstract)
[8]	P.S. Aleksandrov, "Some results in the theory of topological spaces, obtained within the last twenty-five years" Russian Math. Surveys , 15 : 2 (1960) pp. 23–83 Uspekhi Mat. Nauk , 15 : 2 (1960) pp. 25–95
[9]	H. Wallman, "Separation spaces" Ann. of Math. (2) , 42 : 3 (1941) pp. 687–697
[10]	V.I. [V.I. Zaitsev] Zaicev, "Projection spectra" Trans. Moscow Math. Soc. , 27 (1972) pp. 135–200 Trudy Moskov. Mat. Obshch. , 27 (1972) pp. 129–193
[11]	I.M. Gel'fand, D.A. Raikov, G.E. Shilov, "Commutative normed rings" Uspekhi Mat. Nauk , 1 : 2 (1946) pp. 48–146 (In Russian)
[12]	P. Samuel, "Ultrafilters and compactifications of uniform spaces" Trans. Amer. Math. Soc. , 64 (1948) pp. 100–132
[13]	A.V. Arkhangel'skii, "$Q$-compactifications of metric spaces" Math. USSR-Sb. , 20 : 1 (1973) pp. 85–94 Mat. Sb. , 91 (132) : 1 (1973) pp. 78–87
[14]	V. Malykhin, "On countable spaces having no bicompactification of countable tightness" Soviet Math. Dokl. , 13 : 5 (1972) pp. 1407–1411 Dokl. Akad. Nauk SSSR , 206 : 6 (1972) pp. 1293–1296
[15a]	P.S. Aleksandrov, "Some basic directions in general topology" Russian Math. Surveys , 19 : 6 (1964) pp. 1–40 Uspekhi Mat. Nauk , 19 : 6 (1964) pp. 3–46
[15b]	P.S. Aleksandrov, "Corrections to "Some basic directions in general topology" " Russian Math. Surveys , 20 : 1 (1965) pp. 177 Uspekhi Mat. Nauk , 20 : 1 (1965) pp. 253–254
[16]	P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)
[17a]	Yu.M. Smirnov, "On the dimension of remainders of compact Hausdorff extensions of proximity and topological spaces" Mat. Sb. , 69 : 1 (1966) pp. 141–160 (In Russian)
[17b]	Yu.M. Smirnov, "On the dimension of remainders of compact Hausdorff extensions of proximity and topological spaces" Mat. Sb. , 71 : 4 (1966) pp. 454–482 (In Russian)
[18]	K.D. Magill, jr, "The lattice of compactifications of a locally compact group" Proc. London Math. Soc. (3) , 18 (1968) pp. 231–244
[19]	V.V. Fedorchuk, "Perfect irreducible mappings and generalized proximities" Math. USSR-Sb. , 5 : 4 (1968) pp. 489–508 Mat. Sb. , 76 (118) : 4 (1968) pp. 513–536
[20]	H. Freudenthal, "Neuaufbau der Endentheorie" Ann. of Math. (2) , 43 : 2 (1942) pp. 261–279
[21]	Yu.M. Smirnov, "An instance of a one-dimensional normal space contained in no one-dimensional bicompact space" Dokl. Akad. Nauk SSSR , 117 : 6 (1957) pp. 939–942 (In Russian)
[22]	V.M. Ul'yanov, "Solution of a basic problem on compactifications of Wallman type" Soviet Math. Dokl. , 18 (1977) pp. 567–571 Dokl. Akad. Nauk SSSR , 233 : 6 (1977) pp. 1056–1059

Comments

A space is called punctiform if no compact connected subset of it has more than one point. A centred system of closed sets is a collection of closed sets such that any finite intersection is non-empty; such a family is also called a filtered system of sets or simply a filter.

References

[a1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)