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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
2. Nagata, J., Modern general topology, (Wiley Interscience, New York, 1968).Google Scholar