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Spaces which Cannot be Written as a Countable Disjoint Union of Closed Subsets

Published online by Cambridge University Press:  20 November 2018

C. Eberhart ,
J. B. Fugate  and
L. Mohler
C. Eberhart
Affiliation:
University of Kentucky, Lexington, Kentucky
J. B. Fugate
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
L. Mohler
Affiliation:
State University of New york, Buffalo, New york
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It is well known (see [3](1)) that no continuum (i.e. compact, connected, Hausdorff space) can be written as a countable disjoint union of its (nonvoid) closed subsets. This result can be generalized in two ways into the setting of locally compact, connected, Hausdorff spaces. Using the one point compactification of a locally compact, connected, Hausdorff space X one can easily show that X cannot be written as a countable disjoint union of compact subsets. If one makes the further assumption that X is locally connected, then one can show that X cannot be written as a countable disjoint union of closed subsets.(2)

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 3 , September 1973 , pp. 435 - 437
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Knaster, B., Lelek, A., and Mycielski, Jan, Sur les décompositions d’ensembles connexes, Colloq. Math. VI (1958), 227–249.Google Scholar
2. Moore, R. L., Foundations of point set theory, Colloq. Publ., Vol. XIII (revised edition), Amer. Math. Soc, Providence, R.I., 1962.Google Scholar
3. Sierpinski, W., Un théorème sur les ensembles fermés, Bull, de l’Académie des Sciences, Cracovie, (1918), 49–51.Google Scholar
4. Willard, S., General topology, Addison-Wesley, Reading, Mass., 1970.Google Scholar

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