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Compact Subsets in Function Spaces

Published online by Cambridge University Press:  20 November 2018

S.K. Kaul
S.K. Kaul*
Affiliation:
University of Saskatchewan, Regina
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We wish to study the problem of finding conditions under which a family of maps from one space into another, with a suitable topology, is compact. Some of the results obtained in this direction are in [1; 2; 3]. We propose to give conditions, to be called uniformly regular and regular (the terminology is motivated by [4]), under which "Ascoli" theorems can be proved. These notions turn out to be equivalent to even continuity of Kelley [1, page 235] under such conditions that all the theorems in the section on even continuity in it still hold when in their statements even continuity is replaced by either uniform regularity or regularity (see Theorem A below).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 4 , August 1969 , pp. 461 - 466
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Kelley, J. L., General topology. (Van Nostrand, 1955.)Google Scholar
2. Gale, D., Compact sets of functions and function rings. Proc. Amer. Math. Soc. 1 (1950) 303–308.Google Scholar
3. Grothendieck, A., Critères de compacité dans les espaces fonctionnels généraux. Amer. J. Math. 74 (1952) 168–186.Google Scholar
4. Kerekjarto, B.V., Topologische charakterisierung der linearen Abbildungen. Acta. Litt. ac. Sci. Szeged 6 (1934), 235–262.Google Scholar
5. Pontrjagin, L., Topological groups. (Princeton Univ. Press, 1958.)Google Scholar
6. Simmons, G. F., Introduction to topology and modern analysis. (McGraw-Hill Book Co., 1963.)Google Scholar