Supports of continuous functions
Author:
Mark Mandelker
Journal:
Trans. Amer. Math. Soc. 156 (1971), 7383
MSC:
Primary 54.52
MathSciNet review:
0275367
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 [C]
W.
W. Comfort, On the Hewitt realcompactification of
a product space, Trans. Amer. Math. Soc. 131 (1968), 107–118.
MR
0222846 (36 #5896), http://dx.doi.org/10.1090/S00029947196802228461
 [C]
W.
W. Comfort, A theorem of StoneČech type, and a theorem of
Tychonoff type, without the axiom of choice; and their realcompact
analogues, Fund. Math. 63 (1968), 97–110. MR 0236880
(38 #5174)
 [D]
R. F. Dickman, Jr., Compactifications and realcompactifications of arbitrary topological spaces (to appear).
 [F]
Zdeněk
Frolík, Applications to complete families of continuous
functions to the theory of 𝑄spaces, Czechoslovak Math. J.
11 (86) (1961), 115–133 (English, with Russian
summary). MR
0126828 (23 #A4122)
 [GJ]
Leonard
Gillman and Meyer
Jerison, Rings of continuous functions, The University Series
in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton,
N.J.TorontoLondonNew York, 1960. MR 0116199
(22 #6994)
 [H]
A. W. Hager, On the tensor product of function rings, Doctoral Dissertation, Pennsylvania State University, University Park, Pa., 1965.
 [H]
Douglas
Harris, An internal characterization of realcompactness,
Canad. J. Math. 23 (1971), 439–444. MR 0290333
(44 #7517)
 [H]
Allan
Hayes, Alexander’s theorem for realcompactness, Proc.
Cambridge Philos. Soc. 64 (1968), 41–43. MR 0221472
(36 #4524)
 [H]
Edwin
Hewitt, On two problems of Urysohn, Ann. of Math. (2)
47 (1946), 503–509. MR 0017527
(8,165g)
 [H]
Edwin
Hewitt, Rings of realvalued continuous
functions. I, Trans. Amer. Math. Soc. 64 (1948), 45–99. MR 0026239
(10,126e), http://dx.doi.org/10.1090/S00029947194800262399
 [K]
Irving
Kaplansky, Topological rings, Amer. J. Math.
69 (1947), 153–183. MR 0019596
(8,434b)
 [K]
C.
W. Kohls, Ideals in rings of continuous functions, Fund. Math.
45 (1957), 28–50. MR 0102731
(21 #1517)
 [M]
M.
Mandelker, Round 𝑧filters and round subsets of
𝛽𝑋, Israel J. Math. 7 (1969),
1–8. MR
0244951 (39 #6264)
 [N]
Stelios
Negrepontis, Baire sets in topological spaces, Arch. Math.
(Basel) 18 (1967), 603–608. MR 0220248
(36 #3314)
 [N]
N.
Noble, Ascoli theorems and the exponential
map, Trans. Amer. Math. Soc. 143 (1969), 393–411. MR 0248727
(40 #1978), http://dx.doi.org/10.1090/S00029947196902487276
 [P]
Donald
Plank, On a class of subalgebras of 𝐶(𝑋) with
applications to 𝛽𝑋𝑋, Fund. Math.
64 (1969), 41–54. MR 0244953
(39 #6266)
 [R]
Stewart
M. Robinson, The intersection of the free maximal
ideals in a complete space, Proc. Amer. Math.
Soc. 17 (1966),
468–469. MR 0188974
(32 #6401), http://dx.doi.org/10.1090/S00029939196601889748
 [R]
Stewart
M. Robinson, A note on the intersection of free maximal
ideals, J. Austral. Math. Soc. 10 (1969),
204–206. MR 0246132
(39 #7438)
 [W]
M. D. Weir, A net characterization of realcompactness (to appear).
 [C]
 W. W. Comfort, On the Hewitt realcompactification of a product space, Trans. Amer. Math. Soc. 131 (1968), 107118. MR 36 #5896. MR 0222846 (36:5896)
 [C]
 , A theorem of StoneČech type, and a theorem of Tychonoff type, without the axiom of choice; and their realcompact analogues, Fund. Math. 63 (1968), 97110. MR 38 #5174. MR 0236880 (38:5174)
 [D]
 R. F. Dickman, Jr., Compactifications and realcompactifications of arbitrary topological spaces (to appear).
 [F]
 Z. Frolik, Applications of complete families of continuous functions to the theory of Qspaces, Czechoslovak Math. J. 11 (86) (1961), 115133. MR 23 #A4122. MR 0126828 (23:A4122)
 [GJ]
 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)
 [H]
 A. W. Hager, On the tensor product of function rings, Doctoral Dissertation, Pennsylvania State University, University Park, Pa., 1965.
 [H]
 D. Harris, An internal characterization of realcompactness, Canad. J. Math. (to appear). MR 0290333 (44:7517)
 [H]
 A. Hayes, Alexander's theorem for realcompactness, Proc. Cambridge Philos. Soc. 64 (1968), 4143. MR 36 #4524. MR 0221472 (36:4524)
 [H]
 E. Hewitt, On two problems of Urysohn, Ann. of Math. (2) 47 (1946), 503509. MR 8, 165. MR 0017527 (8:165g)
 [H]
 , Rings of realvalued continuous functions. I, Trans. Amer. Math. Soc. 64 (1948), 4599. MR 10, 126. MR 0026239 (10:126e)
 [K]
 I. Kaplansky, Topological rings, Amer. J. Math. 69 (1947), 153183. MR 8, 434. MR 0019596 (8:434b)
 [K]
 C. W. Kohls, Ideals in rings of continuous functions, Fund. Math. 45 (1957), 2850. MR 21 #1517. MR 0102731 (21:1517)
 [M]
 M. Mandelker, Round zfilters and round subsets of , Israel J. Math. 7 (1969), 18. MR 39 #6264. MR 0244951 (39:6264)
 [N]
 S. Negrepontis, Baire sets in topological spaces, Arch. Math. (Basel) 18 (1967), 603608. MR 36 #3314. MR 0220248 (36:3314)
 [N]
 N. Noble, Ascoli theorems and the exponential map, Trans. Amer. Math. Soc. 143 (1969), 393412. MR 40 #1978. MR 0248727 (40:1978)
 [P]
 D. Plank, On a class of subalgebras of with applications to , Fund. Math. 64 (1969), 4154. MR 39 #6266. MR 0244953 (39:6266)
 [R]
 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)
 [R]
 , A note on the intersection of free maximal ideals, J. Austral. Math. Soc. 10 (1969), 204206. MR 39 #7438. MR 0246132 (39:7438)
 [W]
 M. D. Weir, A net characterization of realcompactness (to appear).
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
54.52
Retrieve articles in all journals
with MSC:
54.52
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
