Vol. 62, No. 2, 1976

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On the cardinality relationships between discrete collections and open covers

James Robert Boone

Vol. 62 (1976), No. 2, 359–364
Abstract

This paper unifies and extends various theorems which deal with the relationship between the cardinality of discrete collections and the cardinality of open coverings. For this purpose, the class of spaces which are irreducible of order α is defined. This class includes the δ𝜃-refinable and the [α,)-refinable spaces. Some examples of applications of this class are: a space is [α,)-compact if and only if it is irreducible of order α and has the α BW property and if X is irreducible of order Δ(X), then Δ(X) L(X) Δ(X)+. The open ordinal space [0,Ω) serves as a model to generate examples of spaces which are irreducible of order α but not irreducible of order β, if β > α.

Mathematical Subject Classification 2000
Primary: 54D20
Secondary: 54F45
Milestones
Received: 8 May 1975
Revised: 12 November 1975
Published: 1 February 1976
Authors
James Robert Boone