Compactifications and Semi-Normal Spaces

Frink, Orrin

doi:10.2307/2373025

Cited by 127 publications

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“…(ii) if A ∈ D and x ∈ X \ A, then there exists B ∈ D such that x ∈ B ⊆ X \ A; [6]). Let us mention that V. M. Ul'yanov gave in [14] a solution to the famous problem of O. Frink on Wallman compactifications (cf.…”

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A little more on the product of two pseudocompact spaces

Wajch

1996

Colloq. Math.

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( LÓDŹ) 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 on X ×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: All the spaces considered below are assumed to be completely regular and Hausdorff. As usual, the symbol C(X) will stand for the algebra of continuous real functions defined on X, and C * (X) for the subalgebra of C(X) consisting of bounded functions.

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“…Let us mention that V. M. Ul'yanov gave in [14] a solution to the famous problem of O. Frink on Wallman compactifications (cf. [6]) by proving that a compactification of a Tikhonov space need not be of Wallman type.…”

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A little more on the product of two pseudocompact spaces

Wajch

1996

Colloq. Math.

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AbstractBiles (1970) has called a subring A of the ring C(X), of all real valued continuous functions on a topological space X, a Wallman ring on X whenever Z(A), the zero sets of functions belonging to A, forms a normal base on A'in the sense of Frink (1964). Previously, we have related algebraic properties of a Wallman ring A to topological properties of the Wallman compactification w(Z(A)) of X determined by the normal base Z{A).

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On generalizations of C*-embedding for wallman rings

Bentley

Taylor

1978

J. Aust. Math. Soc.

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Biles (1970) has called a subring A of the ring C(X), of all real valued continuous functions on a topological space X, a Wallman ring on X whenever Z(A), the zero sets of functions belonging to A, forms a normal base on A'in the sense of Frink (1964). Previously, we have related algebraic properties of a Wallman ring A to topological properties of the Wallman compactification w(Z(A)) of X determined by the normal base Z{A). Here we introduce two different generalizations of the concept of "a C*-embedded subset" and study relationships between these and topological (respectively, algebraic) properties of w(Z(A)) (respectively, A).

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“…It is the F-compactification of X where F is a two-point set [10]. It is also an example of a Wallman compactification, introduced in [11] and generalized in [13].…”

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