Extensions and absolutes of hausdorff spaces

Compactness and compactifications in generalized topology

This survey paper is devoted to the algebraic theory of rings of continuous functions. The author considers the relationship between topological spaces and the corresponding rings of continuous functions. The ring properties of rings of continuous functions are discussed. The general theory of sheaves of rings is considered briefly. The importance of rings of functions in mathematics is demonstrated.

Rings of continuous functions. Algebraic aspects

0. Introduction. All the spaces considered below are assumed to be completely regular and Hausdorff. For a space X, denote by K(X) the family of all compactifications of X; βX stands for the Čech–Stone compactification. If αX ∈ K(X), let Cα(X) stand for the set of those functions f ∈ C∗(X) which are continuously extendable over αX. For f ∈ Cα(X), let f be the continuous extension of f over αX and, for F ⊂ Cα(X), let F = {f : f ∈ F}. Suppose that F ⊂ C∗(X). Define ZF (X) as the family of all sets of the form ⋂∞ i=1 ⋃ni j=1 f −1 i,j ([ai,j ; bi,j ]) where fi,j ∈ F and ai,j ≤ bi,j (ai,j , bi,j ∈ R) for i ∈ N and j = 1, . . . , ni (ni ∈ N). Denote by BF (X) the smallest σ-algebra of subsets of X, containing ZF (X). Let SF (X) stand for the collection of all sets that are obtained from ZF (X) by the Souslin operation (cf. [11]). For αX ∈ K(X), put Zα(X) = ZF (X), Bα(X) = BF (X) and Sα(X) = SF (X) with F = Cα(X). Let E(X) be the family of all F ⊂ C∗(X) such that the diagonal mapping eF = ∆f∈F f is a homeomorphic embedding. If F ∈ E(X), then the closure of eF (X) in R|F | is a compactification of X called generated by F and denoted by eF X. By a slight modification of the proof of Theorem 6 of [13] we get

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Pseudocompactness - from compactifications to multiplication of borel sets

We improv the e intuition and some idea osf G. D Faulkne. r and M. C Viper. apresented in the article "Remainders of compactifications and their relation to aquotient lattice of the topology" [Proc. Amer. Math. Soc. 122 (1994) i]n connec-tion with the question about internal condition for localls y compact space Xs andY under whic \h X /3X = f)Y \ Y or, more generally, under whic the remainderh sof compactification osf X belon tog the collection of the remainder osf compacti-fications of Y. We poin out t the reason why the quotient lattic oef the topologyconsidered by Faulkner and Vipera cannot lea tod a satisfactory answe to therabove question. We replace their lattic by thee qoutien t lattic oef a new equiva-lence relation o n a Wallman bas ien order to describe a method of constructing aWallman-type compactification which allow tos deduc us e more complete solutionsto the problems investigate byd Faulkne r and Vipera.

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Compactifications and quotient lattices of wallman bases

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Complete rings of functions and Wallman-Frink compactifications

0. Introduction. The main aim of this paper is to answer the question of when βX × βY is the Wallman compactification of X × Y with respect to the normal base consisting of the zero-sets of all those continuous real functions defined onX×Y which are continuously extendable over βX×βY . In passing, we shall obtain several new conditions which are necessary and sufficient for X × Y to be pseudocompact. To provide a framework for our discussion, let us recall that a normal base D for a Tikhonov space X is a base for the closed sets of X which is stable under finite unions and finite intersections and has the following properties:

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Eliza Wajch

Papers

Complete rings of functions and Wallman-Frink compactifications

Pseudocompactness - from compactifications to multiplication of borel sets

A little more on the product of two pseudocompact spaces