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Countable Compagtifications

Published online by Cambridge University Press:  20 November 2018

Kenneth D. Magill Jr.
Kenneth D. Magill Jr.*
Affiliation:
State University of New York at Buffalo
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It is assumed that all topological spaces discussed in this paper are Hausdorff. By a compactification αX of a space X we mean a compact space containing X as a dense subspace. If, for some positive integer n, αX — X consists of n points, we refer to αX as an n-point compactification of X, in which case we use the notation αn X. If αX — X is countable, we refer to αX as a countable compactification of X. In this paper, the statement that a set is countable means that its elements are in one-to-one correspondence with the natural numbers. In particular, finite sets are not regarded as being countable. Those spaces with n-point compactifications were characterized in (3). From the results obtained there it followed that the only n-point compactifications of the real line are the well-known 1- and 2-point compactifications and the only n-point compactification of the Euclidean N-space, EN (N > 1), is the 1-point compactification.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 616 - 620
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Gillman, L. and Jerison, M., Rings of continuous functions (New York, 1960).Google Scholar
2. Kelley, J. L., General topology (New York, 1955).Google Scholar
3. Magill, K. D. Jr., N-point compactifications. Amer. Math. Monthly 72 (1965), 1075–1081.Google Scholar