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Concerning σ-Connectedness of Baire Spaces

Published online by Cambridge University Press:  20 November 2018

A. García-Máynez
A. García-Máynez*
Affiliation:
Universidad Nacional Autonoma de Mexico, Mexico
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A well known theorem of Sierpiński states that every compact connected Hausdorff space is σ-connected. Hence, if X is locally compact and Hausdorff and X is locally connected at x, then x has a σ-connected neighborhood. However, local connectedness at x is not a necessary condition for x to have a σ-connected neighborhood, because the whole space may be σ-connected without being locally connected at x. One of the purposes of the present paper is then to investigate which points of a given locally compact Hausdorff space have σ-connected neighborhoods. We find also sufficient conditions for a connected, hereditarily Baire space to be σ-connected and prove the impossibility of expressing a connected, Čech-complete, rim compact space as a countable infinite union of mutually disjoint compact sets. Finally, we introduce the concept of D-connected space and relate it to σ-connectedness.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1438 - 1447
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Bourbaki, N., Topologie générale (Hermann, Paris, 1961).Google Scholar
2. García-Máynez, A., On a-connected sets, Bol. Soc. Mat. Mex. 16 (1971), 46–51.Google Scholar
3. Isbell, J. R., Uniform spaces, Mathematical Surveys of the Amer. Math. Soc. 12 (1964).CrossRefGoogle Scholar
4. Knaster, B., Lelek, A. and Mycielski, J., Sur les decompositions d'ensembles connexes, Coll. Math. 6 (1958), 227–246.Google Scholar
5. Mazurkiewicz, S., Sur les continus plans non bornes, Fund. Math. 5 (1924), 188–195.Google Scholar
6. Wilder, R. L., Topology of manifolds, Amer. Math. Soc. Coll. Publ. 32 (New York, 1949).Google Scholar