Displaying similar documents to “Nowhere dense subsets and Booth's Lemma”

Topological characterization of the small cardinal i

Antonio de Padua Franco-Filho (2003)

Commentationes Mathematicae Universitatis Carolinae

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We show that the small cardinal number i = min { | 𝒜 | : 𝒜 is a maximal independent family} has the following topological characterization: i = min { κ ≤ c : { 0 , 1 } κ has a dense irresolvable countable subspace}, where { 0 , 1 } κ denotes the Cantor cube of weight κ . As a consequence of this result, we have that the Cantor cube of weight c has a dense countable submaximal subspace, if we assume (ZFC plus i = c ), or if we work in the Bell-Kunen model, where i = ℵ 1 and c = ℵ ω 1 .

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...

𝒫 -approximable compact spaces

Mihail G. Tkachenko (1991)

Commentationes Mathematicae Universitatis Carolinae

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For every topological property 𝒫 , we define the class of 𝒫 -approximable spaces which consists of spaces X having a countable closed cover γ such that the “section” X ( x , γ ) = ⋂ { F ∈ γ : x ∈ F } has the property 𝒫 for each x ∈ X . It is shown that every 𝒫 -approximable compact space has 𝒫 , if 𝒫 is one of the following properties: countable tightness, ℵ 0 -scatteredness with respect to character, C -closedness, sequentiality (the last holds under MA or 2 ℵ 0 < 2 ℵ 1 ). Metrizable-approximable spaces are studied: every compact space in...