Resolvability and monotone normality

Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán

doi:10.1007/s11856-008-1017-y

Cited by 9 publications

(8 citation statements)

References 9 publications

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“…Let F = G ∩ V 1 and consider the space X = X(F ). As it was observed in [6], spaces obtained as X(H) from some filter H are monotonically normal and ωresolvable.…”

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confidence: 63%

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Resolvability in c.c.c. generic extensions

Soukup¹,

Stanley²

2018

Commentationes Mathematicae Universitatis Carolinae

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“…Let F = G ∩ V 1 and consider the space X = X(F ). As it was observed in [6], spaces obtained as X(H) from some filter H are monotonically normal and ωresolvable.…”

mentioning

confidence: 63%

“…The space X = X(U) is monotonically normal by [6,Theorem 3.1]. An ultrafilter U is λ-descendingly complete if {U ζ : ζ < λ} = ∅ for each decreasing sequence {U ζ : ζ < λ} ⊂ U.…”

mentioning

confidence: 99%

Resolvability in c.c.c. generic extensions

Soukup¹,

Stanley²

2018

Commentationes Mathematicae Universitatis Carolinae

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“…Construction. We use the following, somewhat different topology, inspired by filtration spaces [6] and resolutions [13].…”

Section: Non-reconstructible Spaces and Propertiesmentioning

confidence: 99%

A Topological Variation of the Reconstruction Conjecture

Pitz

Suabedissen

2016

Glasgow Math. J.

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This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space X which are obtained by deleting singletons determine X uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible.We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals R, the rationals Q and the irrationals P are reconstructible, as well as spaces occurring as Stone-Čech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set C.

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“…Indeed, this space is countable, and hence it is trivially a σ-space. Moreover, it is monotonically normal by Theorem 3.2 of [39]. If F is a Ramsey ultrafilter (which exists, for example, if one assumes CH), then Seq(F ) is even a topological group (see [63]).…”

Section: D-separabilitymentioning

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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Resolvability and monotone normality

Cited by 9 publications

References 9 publications

Resolvability in c.c.c. generic extensions

Resolvability in c.c.c. generic extensions

A Topological Variation of the Reconstruction Conjecture

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

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