Monotonically countably paracompact, collectionwise Hausdorff spaces and measurable cardinals
Authors:
Chris Good and Robin W. Knight
Journal:
Proc. Amer. Math. Soc. 134 (2006), 591597
MSC (2000):
Primary 54C10, 54D15, 54D20, 54E20, 54E30
Published electronically:
June 14, 2005
MathSciNet review:
2176028
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We show that, if an MCP (monotonically countably paracompact) space fails to be collectionwise Hausdorff, then there is a measurable cardinal and that, if there are two measurable cardinals, then there is an MCP space that fails to be collectionwise Hausdorff.
 1.
Dennis
K. Burke, PMEA and first countable, countably
paracompact spaces, Proc. Amer. Math. Soc.
92 (1984), no. 3,
455–460. MR
759673 (85h:54032), http://dx.doi.org/10.1090/S00029939198407596737
 2.
W.
W. Comfort and S.
Negrepontis, The theory of ultrafilters, SpringerVerlag, New
YorkHeidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften,
Band 211. MR
0396267 (53 #135)
 3.
Chris
Good, Robin
Knight, and Ian
Stares, Monotone countable paracompactness, Topology Appl.
101 (2000), no. 3, 281–298. MR 1733809
(2000k:54024), http://dx.doi.org/10.1016/S01668641(98)00128X
 4.
C.
Good, D.
W. McIntyre, and W.
S. Watson, Measurable cardinals and finite intervals between
regular topologies, Topology Appl. 123 (2002),
no. 3, 429–441. MR 1924043
(2003h:54001), http://dx.doi.org/10.1016/S01668641(01)002103
 5.
Ge
Ying and Chris
Good, A note on monotone countable paracompactness, Comment.
Math. Univ. Carolin. 42 (2001), no. 4, 771–778.
MR
1883385 (2003a:54027)
 6.
Ryszard
Engelking, General topology, 2nd ed., Sigma Series in Pure
Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from
the Polish by the author. MR 1039321
(91c:54001)
 7.
William
G. Fleissner, The normal Moore space conjecture and large
cardinals, Handbook of settheoretic topology, NorthHolland,
Amsterdam, 1984, pp. 733–760. MR 776635
(86m:54023)
 8.
Gary
Gruenhage, Generalized metric spaces, Handbook of
settheoretic topology, NorthHolland, Amsterdam, 1984,
pp. 423–501. MR 776629
(86h:54038)
 9.
R.
E. Hodel, Spaces characterized by sequences of covers which
guarantee that certain sequences have cluster points, Proceedings of
the University of Houston Point Set Topology Conference (Houston, Tex.,
1971) Univ. Houston, Houston, Tex., 1971, pp. 105–114. MR 0407810
(53 #11580)
 10.
Kenneth
Kunen, Set theory, Studies in Logic and the Foundations of
Mathematics, vol. 102, NorthHolland Publishing Co., Amsterdam, 1983.
An introduction to independence proofs; Reprint of the 1980 original. MR 756630
(85e:03003)
 11.
A.
Lévy and R.
M. Solovay, Measurable cardinals and the continuum hypothesis,
Israel J. Math. 5 (1967), 234–248. MR 0224458
(37 #57)
 12.
Teodor
C. Przymusiński, Products of normal spaces, Handbook of
settheoretic topology, NorthHolland, Amsterdam, 1984,
pp. 781–826. MR 776637
(86c:54007)
 13.
Mary
Ellen Rudin, Dowker spaces, Handbook of settheoretic
topology, NorthHolland, Amsterdam, 1984, pp. 761–780. MR 776636
(86c:54018)
 14.
W.
Stephen Watson, Separation in countably paracompact
spaces, Trans. Amer. Math. Soc.
290 (1985), no. 2,
831–842. MR
792831 (87b:54016), http://dx.doi.org/10.1090/S0002994719850792831X
 1.
 D. Burke, and first countable, countably paracompact spaces, Proc. Amer. Math. Soc., 92 (1984), 455460. MR 0759673 (85h:54032)
 2.
 W.W. Comfort, and S. Negrepontis, The theory of ultrafilters (SpringerVerlag, Berlin, 1974). MR 0396267 (53:135)
 3.
 C. Good, R. W. Knight and I. S. Stares, Monotone Countable Paracompactness, Topol. Appl., 101 (2000), 281298. MR 1733809 (2000k:54024)
 4.
 C. Good, D. W. McIntyre, W. S. Watson, Measurable cardinals and finite intervals between regular topologies, Topol. Appl., 123 (2002), 429441. MR 1924043 (2003h:54001)
 5.
 C. Good and G. Ying, A note on monotone countable paracompactness, Comment. Math. Univ. Carolinae, 42 (2001), 771778. MR 1883385 (2003a:54027)
 6.
 R. Engelking, General Topology, (Heldermann Verlag, Berlin 1989). MR 1039321 (91c:54001)
 7.
 W. G. Fleissner, The normal Moore space conjecture and large cardinals, in Handbook of settheoretic topology, K. Kunen and J. E. Vaughan, eds. (NorthHolland, Amsterdam, 1984). MR 0776635 (86m:54023)
 8.
 G. Gruenhage, Generalized metric spaces, in Handbook of settheoretic topology, K. Kunen and J. E. Vaughan, eds. (NorthHolland, Amsterdam, 1984). MR 0776629 (86h:54038)
 9.
 R. E. Hodel, Spaces defined by sequences of open covers which guarantee that certian sequences have cluster points, Proceedings of the University of Houston Point Set Topology Conference (Houston, Tex., 1971), (1971), 105114. MR 0407810 (53:11580)
 10.
 K. Kunen, Set Theory, An Introduction to Independence Proofs, North Holland, Amsterdam (1983). MR 0756630 (85e:03003)
 11.
 A. Lévy and R. M. Solovay, Measurable cardinals and the continuum hypothesis, Israel J. Math., 5 (1967), 234248. MR 0224458 (37:57)
 12.
 T. C. Przymusinski, Products of normal spaces, in Handbook of settheoretic topology, K. Kunen and J. E. Vaughan, eds. (NorthHolland, Amsterdam, 1984). MR 0776637 (86c:54007)
 13.
 M. E. Rudin, Dowker spaces, in Handbook of settheoretic topology, K. Kunen and J. E. Vaughan, eds. (NorthHolland, Amsterdam, 1984). MR 0776636 (86c:54018)
 14.
 W. S. Watson, Separation in countably paracompact spaces, Trans. Amer. Math. Soc., 290 (1985), 831842. MR 0792831 (87b:54016)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
54C10,
54D15,
54D20,
54E20,
54E30
Retrieve articles in all journals
with MSC (2000):
54C10,
54D15,
54D20,
54E20,
54E30
Additional Information
Chris Good
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, United Kingdom
Email:
c.good@bham.ac.uk
Robin W. Knight
Affiliation:
Mathematical Institute, University of Oxford, 2429 St Giles’, Oxford OX1 3LB, United Kingdom
Email:
knight@maths.ox.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002993905079657
PII:
S 00029939(05)079657
Keywords:
Monotone countable paracompactness,
MCP,
collectionwise Hausdorff,
measurable cardinals
Received by editor(s):
July 30, 2003
Received by editor(s) in revised form:
September 9, 2004
Published electronically:
June 14, 2005
Communicated by:
Alan Dow
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
