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Separators in Continuous Images of Ordered Continua and Hereditarily Locally Connected Continua

Published online by Cambridge University Press:  20 November 2018

J. Grispolakis ,
J. Nikiel ,
J. N. Simone  and
E. D. Tymchatyn
J. Grispolakis
Affiliation:
Technical University of Crete Chania, Crete Greece
J. Nikiel
Affiliation:
Department of Mathematics Texas A & M University College Station, Texas 77843 U.S.A.
J. N. Simone
Affiliation:
Department of Mathematics University of Saskatchewan Saskatoon, Saskatchewan S7N 0W0
E. D. Tymchatyn
Affiliation:
8901 W74th Street #25 Kansas City, Missouri 66204 U.S.A.
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Abstract

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Let X be a Hausdorff space which is the continuous image of an ordered continuum. We prove that every irreducible separator of X is metrizable. This is a far reaching extension of the 1967 theorem of S. Mardešić which asserts that X has a basis of open sets with metrizable boundaries. Our first result is then used to show that, in particular, if Y is an hereditarily locally connected continuum, then for subsets of Y quasi-components coincide with components, and that the boundary of each connected open subset of Y is accessible by ordered continua. These results answer open problems in the literature due to the fourth and third authors, respectively.

Keywords

54F1554C0554F0554F50ordered continuumcontinuous imagehereditarily locally connected continuumcomponentquasi-componentaccessibleirreducible separatortotally disconnected
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 2 , 01 June 1993 , pp. 154 - 163
Copyright
Copyright © Canadian Mathematical Society 1993

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