# Compactification

compact extension

An extension of a topological space which is a compact space. A compactification exists for any topological space, and any -space has a compactification which is a -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 -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 having a Hausdorff compactification of weight .

Two compactifications and of a space are said to be equivalent () if there exists a homeomorphism which is the identity on . Often it is the imbedding itself which is called a compactification. If this definition is accepted, two extensions and are equivalent if there exists a homeomorphism such that . Equivalent compactifications are usually not distinguished, and a class of mutually equivalent compactifications of a space is viewed as a compactification of that space. In such a case one can speak of the set of Hausdorff compactifications of a given (completely-regular) space , since the cardinality of any Hausdorff extension of is at most , while the topologies on a given set also form a set of cardinality .

A compactification follows a compactification () if there exists a continuous mapping which is the identity on . The successor relation converts into a partially ordered set. E. Čech [4] and M.H. Stone [5] showed that the set contains a largest element , the Stone–Čech compactification (or maximal compactification).

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

The maximal compactification is generated by the following proximity :

The Aleksandrov compactification of a locally compact Hausdorff space is generated by the proximity :

The correspondence is an isomorphism between the partially ordered set of proximities on which are compatible with the topology, and the set . The correspondence 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 -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 of weight there exists a set of functions of cardinality such that their diagonal product realizes an imbedding of into the cube (cf. Tikhonov cube). After this, a compactification in of weight is obtained as the closure of in . Čech constructed the maximal compactification of a space using the diagonal product of all continuous functions . 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 can be realized as a space of centred systems of sets open in . 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 from a subordination given on it.

H. Wallman [9] constructed the maximal compactification of a normal space as the space of maximal centred systems of closed sets of this space. The space of maximal centred systems of closed sets of a -space is its -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 let there be given a base of closed sets which is a ring of sets, i.e. contains the intersection and the union of any two elements in it. The base is said to be normal if: 1) for any point and any element not containing this point there exist elements and of the base such that , , ; and 2) for any two elements there exist elements such that , , . 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 , 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 -space is its Wallman compactification , and this bound coincides with the maximal compactification if and only if 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 -compactum makes it possible to describe many properties of completely-regular spaces in terms of properties of -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 -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 to be a space of countable type, i.e. a space in which any -compactum is contained in a -compactum of countable character, it is necessary and sufficient for some (and hence, for all) compactifications that the remainder be finally compact. The spaces of countable type are also interesting because they are normally adjacent to the remainder in all their compactifications , which means that any two non-intersecting sets, closed in the remainder, have neighbourhoods which do not intersect in . As regards imbedding in compactifications, it is finally-compact spaces which are dual with spaces of countable type. A space is finally compact if and only if one (and hence all) of its compactifications have the following property: For any -compactum there exists in the remainder a -compactum containing it, which has countable character in .

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

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

for a perfectly-normal space . One of the first theorems regarding the dimensional properties of compactifications was the theorem according to which any -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 with a countable base have a compactification with zero-dimensional (in the sense of the dimension ind) remainder (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 of weight with , in particular for any normal space of weight with , there exists a compactification of weight and dimension . On the other hand, a completely-regular space is peripherally compact if and only if has a compactification with its remainder zero-dimensionally imbedded in it [8]. The remainder is said to be zero-dimensionally imbedded in (or, relatively zero-dimensional in ) whenever there exists a base of such that

for all , where , the frontier (or boundary) of .

Of obvious importance in the theory of compactifications are perfect compactifications (cf. Perfect compactification) . All perfect compactifications of a space are monotone images of the maximal compactification (in particular, itself is perfect as well) and have, like , a combinatorial structure similar to , but, unlike in the case of , is not always valid, not even for metric spaces . Whereas is the largest perfect compactification, a minimal perfect compactification exists only if has a compactification with a punctiform remainder (in particular, if 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 with a countable base is imbedded in a compactum with remainder of dimension , there exists in an (open) base such that the intersection of the boundaries of any of its elements is compact . This condition is not sufficient for the space to have a compactification with and . Moreover, if is a perfect compactification of , if , and if for any compact set , then for any compactification 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 and have minimal perfect extensions and , any perfect mapping can be extended to yield a mapping .

A topological space of weight is zero-dimensional (i.e. ) if and only if [16] it has a zero-dimensional compactification of weight , so that a space of weight with has a compactification of equal weight and equal dimension. In the case of a completely-regular space with there exists a compactification with and , and this statement is valid for transfinite values of as well ( denotes the weight of ). It follows that a strongly-paracompact metric space has a compactification such that , , and there exists a space such that for all its compactifications in [21].

There are several theorems concerning compactifications of infinite-dimensional spaces. Thus, the maximal compactification of a normal -weakly infinite-dimensional space is weakly infinite-dimensional [16]. Any completely-regular space of weight with a weakly infinite-dimensional compactification (in particular, any normal -weakly infinite-dimensional space of weight ) has a weakly infinite-dimensional compactification of weight . In these theorems it is not allowed to replace the -weak infinite dimension by the -weak infinite dimension (cf. Weakly infinite-dimensional space). Thus, all compactifications of increasing sums of cubes (subsets of the Hilbert cube , 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 is normally adjacent to , where is the (unique) compactification of , then is equal to the smallest of the numbers such that a bordering of multiplicity can be inscribed into each extended bordering (cf. Bordering of a space). A space of countable type has a compactification with a remainder of dimension if and only if a bordering structure of multiplicity , with the basis property, exists in . Moreover, a consequence of the existence of a compactification with a remainder of dimension in a given space is the existence of a compactification of weight with a remainder of dimension .

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

#### References

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