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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Semi-local-connectedness and cut points in metric continua


Author: E. D. Shirley
Journal: Proc. Amer. Math. Soc. 31 (1972), 291-296
MathSciNet review: 0286078
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Abstract: In the first section of this paper, the notion of a space being rational at a point is generalized to what is here called quasi-rational at a point. It is shown that a compact metric continuum which is quasi-rational at each point of a dense subset of an open set is both connected im kleinen and semi-locally-connected on a dense subset of that open set. In the second section a $ {G_\delta }$ set is constructed such that every point in the $ {G_\delta }$ at which the space is not semi-locally-connected is a cut point. A condition is given for this $ {G_\delta }$ set to be dense. This condition, in addition to requiring that the space be not semi-locally-connected at any point of a dense $ {G_\delta }$ set gives a sufficient condition for the space to contain a $ {G_\delta }$ set of cut points. The condition generalizes that given by Grace.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0286078-X
Keywords: Aposyndetic, semi-locally-connected, connected im kleinen, quasi-rational curve, cut point, $ {G_\delta }$ set of cut points
Article copyright: © Copyright 1972 American Mathematical Society