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Screening Properties of the Subbase of all Closed Connected Subsets of a Connectedly Generated Space

Published online by Cambridge University Press:  20 November 2018

J. L. Hursch Jr.
J. L. Hursch Jr.*
Affiliation:
University of Florida, Gainesville, Florida
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In [1] de Groot has introduced the notation “connectedly generated” (or eg) for those spaces in which the closed connected sets form a subbase for the topology. He pointed out that these are the semi-locally connected spaces of Whyburn. See [5; 6].

If X is cg, then, since X is closed, X is the union of a finite number of closed connected sets and, thus, has only a finite number of components. If p is any point in a eg space, and Nv is any neighbourhood of p, then the complement of Nv may be covered by a finite number of closed connected sets, none of which contain p.

In [1] and in [2] the concept of “screening” is introduced and shown to be usefully related to local connectedness and construction of compactifications for completely regular spaces. We review this concept in § 2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 681 - 685
Copyright
Copyright © Canadian Mathematical Society 1970

References

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