Compact Spaces and Compactification: An Algebraic Approach
Hendrik de Vries
Abstract:
Compact Spaces and Compactification
An Algebraic Approach
Hendrik De Vries
It has become a classical result that there exists a complete duality between the theory of boolean algebras and the theory of zero—dimensional compact Hausdorff spaces (M. H. Stone). In this duality, e.g. the maximal proper filters of a boolean algebra correspond (in our approach) to the points of the corresponding Stone space. In this thesis an exposition is given of a theory which deals with a similar algebraization of the theory of arbitrary compact Hausdorff spaces (all topological spaces considered will be Hausdorff spaces). Though a complete duality has been achieved, it has seemed more practicable not to adhere to such a bare duality theory. The notion which supports the whole theory is that of a so—called compingent (boolean) algebra, i.e. a boolean algebra equipped with an additional relation satisfying a certain set of axioms. A typical example of such a compingent algebra is met in the boolean algebra B(C) of all regularly open sets of a compact space C, with the compingent relation “<<” defined by: for a,b \in B(C), a << b \iff \bar{a} \subseteq b. The possibility of a duality theory as indicated was suggested by J. de Groot; only later the close connection with the theory of proximity spaces became apparent to me.
The theory of compingent algebras can also be considered as a topology without points; this approach to this kind of topology seems more promising than that expounded by K. Menger. However, this side of the theory is not further elaborated here.
The points of the compact space attached to a given compingent algebra are obtained as so—calledm aximal concordant filters of the compingent algebra. It appeared that essentially the same filters had been used by P. S. Aleksandrov and H. Freudenthal in more concrete cases for their respective compactification theories. A more general theory of such filters in algebraic structures has been developed by J. G. Horne, mainly in connection with the theory of rings of continuous functions. In this thesis, it is shown that the compactifications of completely regular spaces can be completely described by means of certain compingent algebras. The practicability of this point of view is illuminated by the proofs, in the author’s opinion lucid and simple, of known theorems and generalizations of them. Paradoxically speaking, our method is often more topological than the methods employed in previous proofs.
In the first chapter, the theory of compingent algebras is developed in some detail. For instance, the notion of homomorphism is defined, and its relation to the notion of continuous mapping is studied. Here the relationship with the corresponding Stone theory comes often to the fore.
In the second chapter, the compactification theory of completely regular spaces is developed. The resemblance of the ideas used in this work, and those used by P. S. Aleksandrov, H. Freudenthal , P. Samuel, and J. G. Horne, should be noticed. The fourth section gives briefly the connection of our theory with the theory of proximity spaces as developed by V. A. Efremovič, ]u. M. Smirnov, and A Császár.
It turns out that compingent algebras are also adequate for the description of the continuous mappings of a topological space into compact spaces; the exposition can be found in section 3 of chapter 2.
The last two chapters deal mainly with applications of the previously developed theory. The principal new results are contained in sections 2 and 4 of the third, and section 3 of the fourth chapter.
In chapter 3, § 1, theorems by C. Kuratowski and H. Freudenthal on quasicomponent spaces are generalized. In chapter 3, §2, the notion of percompactness is introduced, being a slight generalization of the notion of peripheral compactness (or semi(bi)compactness). In this more general light, known results on the compactifications of peripherally compact spaces are derived in section 3. The last section of the chapter deals with the problem posed by J. de Groot on the characterization of the complements of n-dimensional sets in compacta. General compact spaces are considered. As main results a sufficient condition is presented and it is shown that the weight of the complement need not necessarily be less than the weight of the compact space.
The first two sections of chapter 4 give generalizations of theorems by E. G. Sklyarenko and C. Kuratowski on weight and dimension preserving compactifications. In the final section, the following two results are proved.
Firstly, if given a set Φ of continuous mappings of a completely regular space T into a compact space D, such that the weight of D and the potency of Φ do not exceed the weight of T, then T can be compactified such that the compactification preserves the weight and the dimension of T and the elements of Φ are continuously extendible to the compactification.
Secondly, if given a completely regular space T and a set Φ of continuous mappings of T into itself whose potency does not exceed the weight of T, then there exists a weight and dimension preserving compactification of T which allows continuous extension of the elements of Φ. Various special cases of this theorem have been proved by several authors, e.g. the result without the condition imposed on the dimensions (J. de Groot and R. H. McDowell).