Partitions of compact Hausdorff spaces

Gary Gruenhage

Fundamenta Mathematicae (1993)

  • Volume: 142, Issue: 1, page 89-100
  • ISSN: 0016-2736

Abstract

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Under the assumption that the real line cannot be covered by ω 1 -many nowhere dense sets, it is shown that (a) no Čech-complete space can be partitioned into ω 1 -many closed nowhere dense sets; (b) no Hausdorff continuum can be partitioned into ω 1 -many closed sets; and (c) no compact Hausdorff space can be partitioned into ω 1 -many closed G δ -sets.

How to cite

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Gruenhage, Gary. "Partitions of compact Hausdorff spaces." Fundamenta Mathematicae 142.1 (1993): 89-100. <http://eudml.org/doc/211974>.

@article{Gruenhage1993,
abstract = {Under the assumption that the real line cannot be covered by $ω_1$-many nowhere dense sets, it is shown that (a) no Čech-complete space can be partitioned into $ω_1$-many closed nowhere dense sets; (b) no Hausdorff continuum can be partitioned into $ω_1$-many closed sets; and (c) no compact Hausdorff space can be partitioned into $ω_1$-many closed $G_δ$-sets.},
author = {Gruenhage, Gary},
journal = {Fundamenta Mathematicae},
keywords = {partitions by closed sets; MA countable},
language = {eng},
number = {1},
pages = {89-100},
title = {Partitions of compact Hausdorff spaces},
url = {http://eudml.org/doc/211974},
volume = {142},
year = {1993},
}

TY - JOUR
AU - Gruenhage, Gary
TI - Partitions of compact Hausdorff spaces
JO - Fundamenta Mathematicae
PY - 1993
VL - 142
IS - 1
SP - 89
EP - 100
AB - Under the assumption that the real line cannot be covered by $ω_1$-many nowhere dense sets, it is shown that (a) no Čech-complete space can be partitioned into $ω_1$-many closed nowhere dense sets; (b) no Hausdorff continuum can be partitioned into $ω_1$-many closed sets; and (c) no compact Hausdorff space can be partitioned into $ω_1$-many closed $G_δ$-sets.
LA - eng
KW - partitions by closed sets; MA countable
UR - http://eudml.org/doc/211974
ER -

References

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  1. [A1] A. V. Arkhangel'skiĭ, On the cardinality of bicompacta satisfying the first axiom of countability, Soviet Math. Dokl. 10 (1969), 951-955. 
  2. [A2] A. V. Arkhangel'skiĭ, Theorems on the cardinality of families of sets in compact Hausdorff spaces, ibid. 17 (1976), 213-217. 
  3. [DP] A. Dow and J. Porter, Cardinalities of H-closed spaces, Topology Proc. 7 (1982), 27-50. Zbl0569.54004
  4. [E] R. Engelking, General Topology, Heldermann, Berlin 1989. 
  5. [FS] D. H. Fremlin and S. Shelah, On partitions of the real line, Israel J. Math. 32 (1979), 299-304. Zbl0413.04002
  6. [K] K. Kunen, Set Theory, North-Holland, Amsterdam 1980. 
  7. [M] A. W. Miller, Covering 2 ω with ω 1 disjoint closed sets, in: The Kleene Symposium, J. Barwise, J. Keisler, and K. Kunen (eds.), North-Holland, 1980, 415-421. 
  8. [N] P. J. Nyikos, A supercompact topology for trees, preprint. 
  9. [S] W. Sierpiński, Un théorème sur les continus, Tôhoku Math. J. 13 (1918), 300-303. Zbl46.0299.03
  10. [SV] P. Štěpánek and P. Vopěnka, Decomposition of metric spaces into nowhere dense sets, Comment. Math. Univ. Carolin. 8 (1967), 387-404. Zbl0179.27803
  11. [T] S. Todorčević, Trees and linearly ordered sets, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, 1984, 235-293. 
  12. [W] S. Watson, Problem Session at the Spring Topology Conference, Univ. of Calif. at Sacramento, April, 1991. 

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