Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The cyclic connectivity of homogeneous arcwise connected continua


Authors: David P. Bellamy and Lewis Lum
Journal: Trans. Amer. Math. Soc. 266 (1981), 389-396
MSC: Primary 54F20
DOI: https://doi.org/10.1090/S0002-9947-1981-0617540-5
MathSciNet review: 617540
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A continuum is cyclicly connected provided each pair of its points lie together on some simple closed curve. In 1927, G. T. Whyburn proved that a locally connected plane continuum is cyclicly connected if and only if it contains no separating points. This theorem was fundamental in his original treatment of cyclic element theory. Since then numerous authors have obtained extensions of Whyburn's theorem. In this paper we characterize cyclic connectedness in the class of all Hausdorff continua.

Theorem. The Hausdorff continuum $ X$ is cyclicly connected if and only if for each point $ x \in X$, $ x$ lies in the relative interior of some arc in $ X$ and $ X - \{ x\} $ is arcwise connected.

We then prove that arcwise connected homogeneous metric continua are cyclicly connected.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54F20

Retrieve articles in all journals with MSC: 54F20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0617540-5
Article copyright: © Copyright 1981 American Mathematical Society