Embeddings into pseudocompact spaces of countable tightness

Bella, Angelo; Pavlov, Oleg

doi:10.1016/j.topol.2003.03.001

Cited by 6 publications

(4 citation statements)

References 5 publications

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“…The idea to use parametrization by ordinal pairs in order to get countable tightness was first suggested by O. Pavlov in the attempt to find a pseudocompact Tychonoff countably tight space without countable fan tightness. This and other results can be found in [4].…”

Section: So We May Suppose Thatsupporting

confidence: 85%

Spaces which are generated by discrete sets

Bella

Šimon

2004

Topology and its Applications

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Section: So We May Suppose Thatsupporting

confidence: 85%

Spaces which are generated by discrete sets

Bella

Šimon

2004

Topology and its Applications

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“…Simon first studied these spaces in [62] (using different terminology) and showed that there are two Whyburn spaces whose product is not weakly Whyburn. In [90], assuming CH, Bella and Simon construct a pseudocompact Whyburn space of countable tightness that is not Fréchet.…”

Section: Convergence Propertiesmentioning

confidence: 99%

Petr Simon (1944-2018)

Hart

Hrušák

Verner

2020

Topology and its Applications

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“…Moving from countably compact to pseudocompact appears much harder. Indeed, with a lot of effort, Bella and Pavlov [20] constructed a Tychonoff pseudocompact space of countable tightness which does not have countable fan tightness. But such a space has a countable set of isolated points and so it is selectively separable.…”

Section: So We Getmentioning

confidence: 99%

Variations of selective separability II: Discrete sets and the influence of convergence and maximality

Bella

Matveev²,

Spadaro

2012

Topology and its Applications

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A space X is called selectively separable (R-separable) if for every sequence of dense subspaces (Dn : n ∈ ω) one can pick finite (respectively, onepoint) subsets Fn ⊂ Dn such that n∈ω Fn is dense in X. These properties are much stronger than separability, but are equivalent to it in the presence of certain convergence properties. For example, we show that every Hausdorff separable radial space is R-separable and note that neither separable sequential nor separable Whyburn spaces have to be selectively separable. A space is called d-separable if it has a dense σ-discrete subspace. We call a space X D-separable if for every sequence of dense subspaces (Dn : n ∈ ω) one can pick discrete subsets Fn ⊂ Dn such that n∈ω Fn is dense in X. Although dseparable spaces are often also D-separable (this is the case, for example, with linearly ordered d-separable or stratifiable spaces), we offer three examples of countable non-D-separable spaces. It is known that d-separability is preserved by arbitrary products, and that for every X, the power X d(X) is d-separable. We show that D-separability is not preserved even by finite products, and that for every infinite X, the power X 2 d(X) is not D-separable. However, for every X there is a Y such that X × Y is D-separable. Finally, we discuss selective and D-separability in the presence of maximality. For example, we show that (assuming d = c) there exists a maximal regular countable selectively separable space, and that (in ZFC) every maximal countable space is D-separable (while some of those are not selectively separable). However, no maximal space satisfies the natural game-theoretic strengthening of D-separability.2000 Mathematics Subject Classification. 54D65, 54A25, 54D55, 54A20.

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Embeddings into pseudocompact spaces of countable tightness

Cited by 6 publications

References 5 publications

Spaces which are generated by discrete sets

Spaces which are generated by discrete sets

Petr Simon (1944-2018)

Variations of selective separability II: Discrete sets and the influence of convergence and maximality

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