Products of compact spaces
Authors:
Victor Saks and R. M. Stephenson
Journal:
Proc. Amer. Math. Soc. 28 (1971), 279288
MSC:
Primary 54.52
MathSciNet review:
0273570
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Abstract: Some results are given on the closure under suitably restricted products of a class of spaces similar to one considered by Z. Frolík and, more recently, by N. Noble. An answer is given to the following question of Gulden, Fleischman, and Weston: Does there exist and an compact space such that some subset of of cardinality is contained in no compact subset of ? It is shown that for every there is a topological group which has this property.
 [1]
Zdeněk
Frolík, Generalisations of compact and Lindelöf
spaces, Czechoslovak Math. J. 9 (84) (1959),
172–217 (Russian, with English summary). MR 0105075
(21 #3821)
 [2]
Zdeněk
Frolík, The topological product of countably compact
spaces, Czechoslovak Math. J. 10 (85) (1960),
329–338 (English, with Russian summary). MR 0117705
(22 #8480)
 [3]
Zdeněk
Frolík, The topological product of two pseudocompact
spaces, Czechoslovak Math. J 10(85) (1960),
339–349 (English, with Russian summary). MR 0116304
(22 #7099)
 [4]
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)
 [5]
Irving
Glicksberg, StoneČech compactifications of
products, Trans. Amer. Math. Soc. 90 (1959), 369–382. MR 0105667
(21 #4405), http://dx.doi.org/10.1090/S00029947195901056674
 [6]
S.
L. Gulden, W.
M. Fleischman, and J.
H. Weston, Linearly ordered topological
spaces, Proc. Amer. Math. Soc. 24 (1970), 197–203. MR 0250272
(40 #3511), http://dx.doi.org/10.1090/S00029939197002502722
 [7]
T. Nakayama, Sets, topology, and algebraic systems, Shibundo, Tokyo, 1949, p. 138.
 [8]
S.
Negrepontis, An example on realcompactifications, Arch. Math.
(Basel) 20 (1969), 162–164. MR 0244952
(39 #6265)
 [9]
Norman
Noble, Countably compact and pseudocompact products,
Czechoslovak Math. J. 19 (94) (1969), 390–397. MR 0248717
(40 #1968)
 [10]
J.
Novák, On the Cartesian product of two compact spaces,
Fund. Math. 40 (1953), 106–112. MR 0060212
(15,640f)
 [11]
Mary
Ellen Rudin, A technique for constructing
examples, Proc. Amer. Math. Soc. 16 (1965), 1320–1323. MR 0188976
(32 #6403), http://dx.doi.org/10.1090/S00029939196501889760
 [12]
C.
T. Scarborough and A.
H. Stone, Products of nearly compact
spaces, Trans. Amer. Math. Soc. 124 (1966), 131–147. MR 0203679
(34 #3528), http://dx.doi.org/10.1090/S00029947196602036797
 [13]
J.
E. Vaughan, Spaces of countable and
pointcountable type, Trans. Amer. Math.
Soc. 151 (1970),
341–351. MR 0266157
(42 #1065), http://dx.doi.org/10.1090/S00029947197002661576
 [14]
R.
Grant Woods, Some ℵ_{𝑂}bounded subsets of
StoneČech compactifications, Israel J. Math.
9 (1971), 250–256. MR 0278266
(43 #3997)
 [1]
 Z. Frolík, Generalisations of compact and Lindelöf spaces, Czechoslovak Math. J. 9 (84) (1959), 172217. MR 21 #3821. MR 0105075 (21:3821)
 [2]
 , The topological product of countably compact spaces, Czechoslovak Math. J. 10 (85) (1960), 329338. MR 22 #8480. MR 0117705 (22:8480)
 [3]
 , The topological product of two pseudocompact spaces, Czechoslovak Math. J. 10 (85) (1960), 339349. MR 22 #7099. MR 0116304 (22:7099)
 [4]
 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)
 [5]
 I. Glicksberg, StoneČech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369382. MR 21 #4405. MR 0105667 (21:4405)
 [6]
 S. L. Gulden, W. M. Fleischman and J. H. Weston, Linearly ordered topological spaces, Proc. Amer. Math. Soc. 24 (1970), 197203. MR 0250272 (40:3511)
 [7]
 T. Nakayama, Sets, topology, and algebraic systems, Shibundo, Tokyo, 1949, p. 138.
 [8]
 S. Negrepontis, An example on realcompactifications, Arch. Math. (Basel) 20 (1969), 162164. MR 39 #6265. MR 0244952 (39:6265)
 [9]
 N. Noble, Countably compact and pseudocompact products, Czechoslovak Math. J. 19 (94) (1969), 390397. MR 40 #1968. MR 0248717 (40:1968)
 [10]
 J. Novák, On the cartesian product of two compact spaces, Fund. Math. 40 (1953), 106112. MR 15, 640. MR 0060212 (15:640f)
 [11]
 M. E. Rudin, A technique for constructing examples, Proc. Amer. Math. Soc. 16 (1965), 13201323. MR 32 #6403. MR 0188976 (32:6403)
 [12]
 C. T. Scarborough and A. H. Stone, Products of nearly compact spaces, Trans. Amer. Math. Soc. 124 (1966), 131147. MR 34 #3528. MR 0203679 (34:3528)
 [13]
 J. E. Vaughan, Spaces of countable and pointcountable type, Trans. Amer. Math. Soc. 151 (1970), 341351. MR 0266157 (42:1065)
 [14]
 R. Grant Woods, Some bounded subsets of StoneČech compactifications, Israel J. Math. (to appear). MR 0278266 (43:3997)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197102735706
PII:
S 00029939(1971)02735706
Keywords:
Product spaces,
countably compact spaces,
compact spaces,
products of compact spaces
Article copyright:
© Copyright 1971
American Mathematical Society
