Displaying similar documents to “Point-countable π-bases in first countable and similar spaces”

In search for Lindelöf C p ’s

Raushan Z. Buzyakova (2004)

Commentationes Mathematicae Universitatis Carolinae

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It is shown that if X is a first-countable countably compact subspace of ordinals then C p ( X ) is Lindelöf. This result is used to construct an example of a countably compact space X such that the extent of C p ( X ) is less than the Lindelöf number of C p ( X ) . This example answers negatively Reznichenko’s question whether Baturov’s theorem holds for countably compact spaces.

On the product of a compact space with an hereditarily absolutely countably compact space

Maddalena Bonanzinga (1997)

Commentationes Mathematicae Universitatis Carolinae

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We show that the product of a compact, sequential T 2 space with an hereditarily absolutely countably compact T 3 space is hereditarily absolutely countably compact, and further that the product of a compact T 2 space of countable tightness with an hereditarily absolutely countably compact ω -bounded T 3 space is hereditarily absolutely countably compact.

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...

Some versions of second countability of metric spaces in ZF and their role to compactness

Kyriakos Keremedis (2018)

Commentationes Mathematicae Universitatis Carolinae

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In the realm of metric spaces we show in ZF that: (i) A metric space is compact if and only if it is countably compact and for every ε > 0 , every cover by open balls of radius ε has a countable subcover. (ii) Every second countable metric space has a countable base consisting of open balls if and only if the axiom of countable choice restricted to subsets of ℝ holds true. (iii) A countably compact metric space is separable if and only if it is second countable.