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Displaying similar documents to “Two results on special points”

Remarks on star countable discrete closed spaces

Yan-Kui Song (2013)

Czechoslovak Mathematical Journal

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In this paper, we prove the following statements: (1) There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable. (2) Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace. (3) Assuming 2 ℵ 0 = 2 ℵ 1 , there exists a normal absolutely star countable discrete closed space having a regular-closed subspace which is not star countable.

Closed discrete subsets of separable spaces and relative versions of normality, countable paracompactness and property ( a )

Samuel Gomes da Silva (2011)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we show that a separable space cannot include closed discrete subsets which have the cardinality of the continuum and satisfy relative versions of any of the following topological properties: normality, countable paracompactness and property ( a ) . It follows that it is consistent that closed discrete subsets of a separable space X which are also relatively normal (relatively countably paracompact, relatively ( a ) ) in X are necessarily countable. There are, however, consistent...

Weak orderability of some spaces which admit a weak selection

Camillo Costantini (2006)

Commentationes Mathematicae Universitatis Carolinae

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We show that if a Hausdorff topological space X satisfies one of the following properties: a) X has a countable, discrete dense subset and X 2 is hereditarily collectionwise Hausdorff; b) X has a discrete dense subset and admits a countable base; then the existence of a (continuous) weak selection on X implies weak orderability. As a special case of either item a) or b), we obtain the result for every separable metrizable space with a discrete dense subset.

Lonely points revisited

Jonathan L. Verner (2013)

Commentationes Mathematicae Universitatis Carolinae

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In our previous paper, we introduced the notion of a lonely point, due to P. Simon. A point p ∈ X is lonely if it is a limit point of a countable dense-in-itself set, it is not a limit point of a countable discrete set and all countable sets whose limit point it is form a filter. We use the space 𝒢 ω from a paper of A. Dow, A.V. Gubbi and A. Szymański [Rigid Stone spaces within ZFC, Proc. Amer. Math. Soc. 102 (1988), no. 3, 745–748] to construct lonely points in ω * . This answers the question...