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Cardinality of discrete subsets of a topological space

Published online by Cambridge University Press:  17 April 2009

David B. Gauld  and
M.K. Vamanamurthy
David B. Gauld
Affiliation:
Department of Mathematics, University of Auckland, Private Bag, Auckland, New Zealand.
M.K. Vamanamurthy
Affiliation:
Department of Mathematics, University of Auckland, Private Bag, Auckland, New Zealand.
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Abstract

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In this paper we prove two results relating the cardinalities of discrete subsets to those of dense subsets of a topological space. In particular, we show that

(1) closed discrete subsets of separable normal spaces are countable, and

(2) discrete subsets of separable regular spaces have cardinalities at most that of the continuum.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 25 , Issue 1 , February 1982 , pp. 99 - 101
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Engelking, Ryszard, General topology (Monografie Matematyczne, 60. PWN – Polish Scientific Publishers, Warszawa, 1977).Google Scholar