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On Σ-Finite Families

Published online by Cambridge University Press:  20 November 2018

Byron H. Mccandless
Byron H. Mccandless*
Affiliation:
Kent State University, Kent, Ohio
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Let be a family of subsets of a topological space X. We do not require to be a covering of X, nor do we assume that the members of are necessarily open. In this paper we shall assume that is of a special sort, which we call Σ-Finite. We show that a Σ-Finite family is both locally finite and star-finite, and in particular that an open covering of X is Σ-Finite if and only if it is star-finite. We then prove that every Σ-Finite family is ᓂ-discrete, so that in particular, every star-finite open covering of X is (ᓂ-discrete. There seems to be some applications of this fact.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 3 , 01 June 1975 , pp. 481 - 488
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Heath, R. W., Screenability, pointwise paracompactness, and metrization of Moore spaces, Can. J. Math. 16 (1964), 763–770.Google Scholar
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