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Search: MSC category 03E15 ( Descriptive set theory [See also 28A05, 54H05] )

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1. CJM Online first

Hrušák, Michael; van Mill, Jan
 Nearly Countable Dense Homogeneous Spaces We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely $n$ types of countable dense sets: such a space contains a subset $S$ of size at most $n{-}1$ such that $S$ is invariant under all homeomorphisms of $X$ and $X\setminus S$ is countable dense homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$ types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or $\mathfrak{c}$ many types of countable dense sets, is shown to be closely related to Topological Vaught's Conjecture. Keywords:countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjectureCategories:54H05, 03E15, 54E50

2. CJM 2012 (vol 64 pp. 1378)

Raghavan, Dilip; Steprāns, Juris
 On Weakly Tight Families Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\mathfrak{c} \lt {\aleph}_{\omega}$, we construct a weakly tight family under the hypothesis $\mathfrak{s} \leq \mathfrak{b} \lt {\aleph}_{\omega}$. The case when $\mathfrak{s} \lt \mathfrak{b}$ is handled in $\mathrm{ZFC}$ and does not require $\mathfrak{b} \lt {\aleph}_{\omega}$, while an additional PCF type hypothesis, which holds when $\mathfrak{b} \lt {\aleph}_{\omega}$ is used to treat the case $\mathfrak{s} = \mathfrak{b}$. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by HruÅ¡Ã¡k and GarcÃ­a Ferreira, who applied it to the KatÃ©tov order on almost disjoint families. Keywords:maximal almost disjoint family, cardinal invariantsCategories:03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65

3. CJM 1999 (vol 51 pp. 309)

Leung, Denny H.; Tang, Wee-Kee
 Symmetric sequence subspaces of $C(\alpha)$, II If $\alpha$ is an ordinal, then the space of all ordinals less than or equal to $\alpha$ is a compact Hausdorff space when endowed with the order topology. Let $C(\alpha)$ be the space of all continuous real-valued functions defined on the ordinal interval $[0, \alpha]$. We characterize the symmetric sequence spaces which embed into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy $(E_\alpha)$ of symmetric sequence spaces is constructed so that, for each countable ordinal $\alpha$, $E_\alpha$ embeds into $C(\omega^{\omega^\alpha})$, but does not embed into $C(\omega^{\omega^\beta})$ for any $\beta < \alpha$. Categories:03E13, 03E15, 46B03, 46B45, 46E15, 54G12