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Displaying similar documents to “Lonely points revisited”

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.

Functional separability

Ronnie Levy, M. Matveev (2010)

Commentationes Mathematicae Universitatis Carolinae

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A space X is functionally countable (FC) if for every continuous f : X → ℝ , | f ( X ) | ≤ ω . The class of FC spaces includes ordinals, some trees, compact scattered spaces, Lindelöf P-spaces, σ -products in 2 κ , and some L-spaces. We consider the following three versions of functional separability: X is 1-FS if it has a dense FC subspace; X is 2-FS if there is a dense subspace Y ⊂ X such that for every continuous f : X → ℝ , | f ( Y ) | ≤ ω ; X is 3-FS if for every continuous f : X → ℝ , there is a dense subspace Y ⊂ X such that | f ( Y ) | ≤ ω . We give examples...

The regular topology on C ( X )

Wolf Iberkleid, Ramiro Lafuente-Rodriguez, Warren Wm. McGovern (2011)

Commentationes Mathematicae Universitatis Carolinae

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Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45–99] defined the m -topology on C ( X ) , denoted C m ( X ) , and demonstrated that certain topological properties of X could be characterized by certain topological properties of C m ( X ) . For example, he showed that X is pseudocompact if and only if C m ( X ) is a metrizable space; in this case the m -topology is precisely the topology of uniform convergence. What is interesting with regards to the m -topology is that it is...