Displaying similar documents to “An example of a space whose all continuous mappings are almost injective”

On relatively almost countably compact subsets

Yan-Kui Song, Shu-Nian Zheng (2010)

Mathematica Bohemica

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A subset Y of a space X is almost countably compact in X if for every countable cover 𝒰 of Y by open subsets of X , there exists a finite subfamily 𝒱 of 𝒰 such that Y ⊆ ⋃ 𝒱 ¯ . In this paper we investigate the relationship between almost countably compact spaces and relatively almost countably compact subsets, and also study various properties of relatively almost countably compact subsets.

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...

On relatively almost Lindelöf subsets

Yankui Song (2009)

Mathematica Bohemica

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A subspace Y of a space X is almost Lindelöf (strongly almost Lindelöf) in X if for every open cover 𝒰 of X (of Y by open subsets of X ), there exists a countable subset 𝒱 of 𝒰 such that Y ⊆ ⋃ { V ¯ V ∈ 𝒱 } . In this paper we investigate the relationships between relatively almost Lindelöf subset and relatively strongly almost Lindelöf subset by giving some examples, and also study various properties of relatively almost Lindelöf subsets and relatively strongly almost Lindelöf subsets.

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.