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Partitions of compact Hausdorff spaces

Gary Gruenhage (1993)

Fundamenta Mathematicae

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.

Planar rational compacta

L. Feggos, S. Iliadis, S. Zafiridou (1995)

Colloquium Mathematicae

In this paper we consider rational subspaces of the plane. A rational space is a space which has a basis of open sets with countable boundaries. In the special case where the boundaries are finite, the space is called rim-finite.

Pressing Down Lemma for λ -trees and its applications

Hui Li, Liang-Xue Peng (2013)

Czechoslovak Mathematical Journal

For any ordinal λ of uncountable cofinality, a λ -tree is a tree T of height λ such that | T α | < cf ( λ ) for each α < λ , where T α = { x ∈ T : ht ( x ) = α } . In this note we get a Pressing Down Lemma for λ -trees and discuss some of its applications. We show that if η is an uncountable ordinal and T is a Hausdorff tree of height η such that | T α | ≤ ω for each α < η , then the tree T is collectionwise Hausdorff if and only if for each antichain C ⊂ T and for each limit ordinal α ≤ η with cf ( α ) > ω , { ht ( c ) : c ∈ C } ∩ α is not stationary in α . In the last part of this note, we investigate some...

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