Supports of continuous functions
Author:
Mark Mandelker
Journal:
Trans. Amer. Math. Soc. 156 (1971), 7383
MSC:
Primary 54.52
MathSciNet review:
0275367
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Abstract: Gillman and Jerison have shown that when X is a realcompact space, every function in that belongs to all the free maximal ideals has compact support. A space with the latter property will be called compact. In this paper we give several characterizations of compact spaces and also introduce and study a related class of spaces, the compact spaces; these are spaces X with the property that every function in with pseudocompact support has compact support. It is shown that every realcompact space is compact and every compact space is compact. A family of subsets of a space X is said to be stable if every function in is bounded on some member of . We show that a completely regular Hausdorff space is realcompact if and only if every stable family of closed subsets with the finite intersection property has nonempty intersection. We adopt this condition as the definition of realcompactness for arbitrary (not necessarily completely regular Hausdorff) spaces, determine some of the properties of these realcompact spaces, and construct a realcompactification of an arbitrary space.
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 L. Gillman and M. Jerison, Rings of continuous functions, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #6994. MR 0116199 (22:6994)
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 A. W. Hager, On the tensor product of function rings, Doctoral Dissertation, Pennsylvania State University, University Park, Pa., 1965.
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 A. Hayes, Alexander's theorem for realcompactness, Proc. Cambridge Philos. Soc. 64 (1968), 4143. MR 36 #4524. MR 0221472 (36:4524)
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 C. W. Kohls, Ideals in rings of continuous functions, Fund. Math. 45 (1957), 2850. MR 21 #1517. MR 0102731 (21:1517)
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 M. Mandelker, Round zfilters and round subsets of , Israel J. Math. 7 (1969), 18. MR 39 #6264. MR 0244951 (39:6264)
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 N. Noble, Ascoli theorems and the exponential map, Trans. Amer. Math. Soc. 143 (1969), 393412. MR 40 #1978. MR 0248727 (40:1978)
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 D. Plank, On a class of subalgebras of with applications to , Fund. Math. 64 (1969), 4154. MR 39 #6266. MR 0244953 (39:6266)
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 S. M. Robinson, The intersection of the free maximal ideals in a complete space, Proc. Amer. Math. Soc. 17 (1966), 468469. MR 32 #6401. MR 0188974 (32:6401)
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 M. D. Weir, A net characterization of realcompactness (to appear).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197102753674
PII:
S 00029947(1971)02753674
Keywords:
Support,
continuous function,
compact support,
pseudocompact support,
free maximal ideal,
realcompact space,
compact space,
compact space,
relatively pseudocompact subset,
stable family,
round subset,
realcompactification
Article copyright:
© Copyright 1971
American Mathematical Society
