Vol. 58, No. 1, 1975

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Wallman rings

Herschel Lamar Bentley and Barbara June Taylor

Vol. 58 (1975), No. 1, 15–35

In 1964 Frink defined a normal base. He hypothesized that every Hausdorff compactification of a Tychonoff space X may be realized as a compactification w() of Wallman type obtained from a normal base on X, where is the family of zero sets for some subring of C(X). Later Biles formally defined a Wallman Ring on a Tychonoff space to be a subring of C(X) whose zero sets form a normal base on X.

The problem in this paper is to study examples of Wallman Rings and develop properties of Wallman Rings. For a locally compact space with a given compactification and a certain type of retract map, a Wallman Ring is defined which induces the given compactification.

General algebraic and topological properties of Wallman Rings are considered. Among the results obtained are “Every Wallman Ring is equivalent to one which contains all rational constant functions” and “An ideal of a Wallman Ring which is itself a Wallman Ring is equivalent to the superring.”

Mathematical Subject Classification 2000
Primary: 54D35
Received: 28 February 1974
Published: 1 May 1975
Herschel Lamar Bentley
Barbara June Taylor