A countable dense homogeneous set of reals of size ℵ₁

Ilijas Farah; Michael Hrušák; Carlos Azarel Martínez Ranero

Displaying similar documents to “A countable dense homogeneous set of reals of size ℵ₁”

Non-meager P-filters are countable dense homogeneous

Rodrigo Hernández-Gutiérrez, Michael Hrušák (2013)

Colloquium Mathematicae

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We prove that if ℱ is a non-meager P-filter, then both ℱ and $^{ω} ℱ$ are countable dense homogeneous spaces.

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 $f : X \to ℝ$ , $| f (X) | \leq ω$ . 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 $Y \subset X$ such that for every continuous $f : X \to ℝ$ , $| f (Y) | \leq ω$ ; $X$ is 3-FS if for every continuous $f : X \to ℝ$ , there is a dense subspace $Y \subset X$ such that $| f (Y) | \leq ω$ . We give examples...

Point-countable π-bases in first countable and similar spaces

V. V. Tkachuk (2005)

Fundamenta Mathematicae

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It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable π-base. We look at general spaces with point-countable π-bases and prove, in particular, that, under the Continuum Hypothesis, any Lindelöf first countable space has a point-countable π-base. We also analyze when the function space $C_{p} (X)$ has a point-countable π -base, giving a criterion for this in terms of the topology of X when l*(X) = ω. Dealing with point-countable π-bases makes...

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.

Three small results on normal first countable linearly H-closed spaces

Mathieu Baillif (2022)

Commentationes Mathematicae Universitatis Carolinae

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We use topological consequences of , and proved by other authors to show that normal first countable linearly H-closed spaces with various additional properties are compact in these models.

Some dimensional results for a class of special homogeneous Moran sets

Xiaomei Hu (2016)

Czechoslovak Mathematical Journal

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We construct a class of special homogeneous Moran sets, called ${m_{k}}$ -quasi homogeneous Cantor sets, and discuss their Hausdorff dimensions. By adjusting the value of ${m_{k}}_{k \geq 1}$ , we constructively prove the intermediate value theorem for the homogeneous Moran set. Moreover, we obtain a sufficient condition for the Hausdorff dimension of homogeneous Moran sets to assume the minimum value, which expands earlier works.

Positively homogeneous functions and the Łojasiewicz gradient inequality

Alain Haraux (2005)

Annales Polonici Mathematici

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It is quite natural to conjecture that a positively homogeneous function with degree d ≥ 2 on $ℝ^{N}$ satisfies the Łojasiewicz gradient inequality with exponent θ = 1/d without any need for an analyticity assumption. We show that this property is true under some additional hypotheses, but not always, even for N = 2.

Besov spaces on spaces of homogeneous type and fractals

Dachun Yang (2003)

Studia Mathematica

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Let Γ be a compact d-set in ℝⁿ with 0 < d ≤ n, which includes various kinds of fractals. The author shows that the Besov spaces $B_{p q}^{s} (Γ)$ defined by two different and equivalent methods, namely, via traces and quarkonial decompositions in the sense of Triebel are the same spaces as those obtained by regarding Γ as a space of homogeneous type when 0 < s < 1, 1 < p < ∞ and 1 ≤ q ≤ ∞.

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.