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Four-Dimension Equivalences

Published online by Cambridge University Press:  20 November 2018

J. R. Gard  and
R. D. Johnson
J. R. Gard
Affiliation:
Georgia Institute of Technology Atlanta, Georgia
R. D. Johnson
Affiliation:
Georgia Institute of Technology Atlanta, Georgia
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The object of this paper is to establish the equivalence of four functionrelated dimension concepts in arbitrary topological spaces. These concepts involve stability of functions (3, p. 74), the modification of covering dimension involving basic covers (1, p. 243) (which is equivalent to Yu. M. Smirnov's definition using normal covers), the definition involving essential mappings (2, p. 496), and a modification of the closed set separation characterization of dimension in (3, p. 35).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 48 - 50
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Gillman, L. and Jerison, M., Rings of continuous functions (Princeton, 1960).10.1007/978-1-4615-7819-2CrossRefGoogle Scholar
2. Hemmingsen, E., Some theorems in dimension theory for normal Hausdorff spaces, Duke Math. J., 13 (1946), 495–504.Google Scholar
3. Hurewicz, W. and Wallman, H., Dimension theory (Princeton, 1948).Google Scholar
4. Smirnov, Yu. M., On the dimension of proximity spaces, Mat. Sb., 38 (80) (1956), 283-302, Amer. Math. Soc. Transi. (2), 21 (1962), 1–20.Google Scholar