Displaying similar documents to “Countable-dimensional spaces: a survey”

Countable dense homogeneity and λ-sets

Rodrigo Hernández-Gutiérrez, Michael Hrušák, Jan van Mill (2014)

Fundamenta Mathematicae

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We show that all sufficiently nice λ-sets are countable dense homogeneous (𝖢𝖣𝖧). From this fact we conclude that for every uncountable cardinal κ ≤ 𝔟 there is a countable dense homogeneous metric space of size κ. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size κ is equivalent to the existence of a λ-set of size κ. On the other hand, it is consistent with the continuum arbitrarily large that every 𝖢𝖣𝖧 metric space has size either ω₁...

Resolving a question of Arkhangel'skiĭ's

Michael G. Charalambous (2006)

Fundamenta Mathematicae

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We construct in ZFC a cosmic space that, despite being the union of countably many metrizable subspaces, has covering dimension equal to 1 and inductive dimensions equal to 2.

On absolutely submetrizable spaces

Raushan Z. Buzyakova (2006)

Commentationes Mathematicae Universitatis Carolinae

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We introduce a notion of absolute submetrizability (= ``every Tychonoff subtopology is submetrizable'') and investigate its behavior under basic topological operations. The main result is an example of an absolutely submetrizable space that contains an uncountable set of isolated points (hence the space is neither separable nor hereditarily Lindelöf). This example is used to show that absolute submetrizability is not preserved by some topological operations, in particular, by free sums. ...