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Failure of normality in the box product of uncountably many real lines
Author(s):
L.
Brian
Lawrence
Journal:
Trans. Amer. Math. Soc.
348
(1996),
187-203.
MSC (1991):
Primary 54D18;
Secondary 54A35, 54B10, 54B20
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Abstract:
We prove in ZFC that the box product of  many copies of  is neither normal nor collectionwise Hausdorff. As an addendum to the proof, we show that if the cardinality of the continuum is  , then these properties also fail in the closed subspace consisting of all functions which assume the value  on all but countably many indices.
References:
- [vD
 ] - E. K. van Douwen, The box product of countably many metrizable spaces need not be normal, Fund. Math. 88 (1975), 127--132. MR 52:6640
- [vD
 ] - ------, Covering and separation properties of box products, Surveys in General Topology, Academic Press, 1980, pp. 55--130. MR 81i:54006
- [Kn]
- C. J. Knight, Box topologies , Quart. J. Math. 15 (1964), 41--54. MR 28:3398
- [Ku
 ] - K. Kunen, Some comments on box products, Colloq. Math. Soc. János Bolyai 10 (1975), 1011--1016. MR 52:15339
- [Ku
 ] - ------, On paracompactness of box products of compact spaces, Trans. Amer. Math. Soc. 240 (1978), 307--316. MR 58:24165
- [L
 ] - L. B. Lawrence, The box product of countably many copies of the rationals is consistently paracompact, Trans. Amer. Math. Soc. 309 (1988), 787--796. MR 89k:54053
- [L
 ] - ------, Toward a theory of normality and paracompactness in box products, Ann. New York Acad. Sci. 705 (1993), 78--91. MR 95a:54038
- [Mic
 ] - E. A. Michael, A note on paracompact spaces, Proc. Amer. Math. Soc. 4 (1953), 831--838. MR 15:144b
- [Mic
 ] - ------, Another note on paracompact spaces, Proc. Amer. Math. Soc. 8 (1957), 822--828. MR 19:299c
- [Mil]
- A. W. Miller, On box products Topology Appl. 14 (1982), 313--317. MR 84a:54014
- [NyPi]
- P. Nyikos and L. Piatkiewicz, Paracompact subspaces in the box product topology, Proc. Amer. Math. Soc. (to appear).
- [Ro
 ] - J. Roitman, Paracompact box products in forcing extensions, Fund. Math. 102 (1978), 219--228.MR 80j:03078
- [Ro
 ] - ------, More paracompact box products, Proc. Amer. Math. Soc. 74 (1979), 171--176. MR 80i:54006
- [Ru
 ] - M. E. Rudin, The box product of countably many compact metric spaces, General Topology and Appl. 2 (1972), 293--298. MR 48:2969
- [Ru
 ] - ------, Lectures on set-theoretic topology, C.B.M.S. Regional Conf. Ser. in Math., no. 23, Amer. Math. Soc., Providence, RI, 1975. MR 51:4128
- [Wil
 ] - S. W. Williams, Is
 paracompact? Topology Proc. 1 (1976), 141--146. MR 58:18347 - [Wil
 ] - ------, Box products, Handbook of Set-Theoretic Topology, (K. Kunen and J. Vaughan, eds.), Elsevier, 1984, pp. 169--200. MR 87a:54007
- [Win
 ] - L. Wingers, Box products of
 -compact spaces, Topology Appl. 56 (1994), 185--197. MR 94m:54026 - [Win
 ] - ------, Box products and Hurewicz spaces, Topology Appl. (to appear).
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Additional Information:
L.
Brian
Lawrence
Affiliation:
Department of Mathematics, George Mason University, Fairfax, Virginia 22030-4444
DOI:
10.1090/S0002-9947-96-01375-X
PII:
S 0002-9947(96)01375-X
Keywords:
Box product,
normal,
paracompact,
collectionwise Hausdorff,
continuum hypothesis
Received by editor(s):
November 22, 1991
Received by editor(s) in revised form:
October 31, 1994
Additional Notes:
An abstract of this paper was presented at the Summer Topology Conference in Honor of Mary Ellen Rudin, University of Wisconsin, Madison, June, 1991
Dedicated:
Dedicated to Mary Ellen Rudin and A. H. Stone
Copyright of article:
Copyright
1996,
American Mathematical Society
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