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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

1. Groot, J. de, Connectedly generated spaces (to appear in Proceedings of the Topological Symposium in Hertceg Novi, Yugoslavia, 1968).Google Scholar
2. Groot, J. de and Aarts, J. M., Complete regularity as a separation axiom,, Can. J. Math. 21 (1969), 96–105.Google Scholar
3. Groot, J. de and McDowell, R. H., Locally connected spaces and their compactifications, Illinois J. Math. 11 (1967), 353–364.Google Scholar
4. Gillman, L. and Jerison, M., Rings of continuous functions, The University Series in Higher Mathematics (Van Nostrand, Princeton, N.J .-Toronto-London-New York, 1960).Google Scholar
5. Whyburn, G. T., Semi-locally connected sets, Amer. J. Math. 61 (1939), 733–749.Google Scholar
6. Whyburn, G. T., Analytic topology, Amer. Math. Soc. Colloq. Publ., Vol. 28 (Amer. Math. Soc, Providence, R.I., 1942, revised, 1955).Google Scholar
7. Wilder, R. L., Topology of manifolds, Amer. Math. Soc. Colloq. Publ., Vol. 32 (Amer. Math. Soc, Providence, R.I., 1949, revised 1963).Google Scholar