Partitions of compact Hausdorff spaces

ArticleOriginal scientific text

Title

Authors ¹

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.

[A1] A. V. Arkhangel'skiĭ, On the cardinality of bicompacta satisfying the first axiom of countability, Soviet Math. Dokl. 10 (1969), 951-955.
[A2] A. V. Arkhangel'skiĭ, Theorems on the cardinality of families of sets in compact Hausdorff spaces, ibid. 17 (1976), 213-217.
[DP] A. Dow and J. Porter, Cardinalities of H-closed spaces, Topology Proc. 7 (1982), 27-50.
[E] R. Engelking, General Topology, Heldermann, Berlin 1989.
[FS] D. H. Fremlin and S. Shelah, On partitions of the real line, Israel J. Math. 32 (1979), 299-304.
[K] K. Kunen, Set Theory, North-Holland, Amsterdam 1980.
[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.
[N] P. J. Nyikos, A supercompact topology for trees, preprint.
[S] W. Sierpiński, Un théorème sur les continus, Tôhoku Math. J. 13 (1918), 300-303.
[SV] P. Štěpánek and P. Vopěnka, Decomposition of metric spaces into nowhere dense sets, Comment. Math. Univ. Carolin. 8 (1967), 387-404.
[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.
[W] S. Watson, Problem Session at the Spring Topology Conference, Univ. of Calif. at Sacramento, April, 1991.

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