On resolvability of topological spaces

Pavlov, Oleg

doi:10.1016/s0166-8641(02)00004-4

Cited by 18 publications

(13 citation statements)

References 7 publications

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“…Actually, this fact is obtained as a corollary of Theorem 2, which states (in one of its possible formulations) that every regular and non-trivial connected space, with a π-base consisting of connected sets, is ω-resolvable. As far as we know, our Theorem 2 and [Pav,Corollary 3.11] are the only results in the literature that show how connectedness and regularity, together with some reasonable supplementary assumptions, may entail resolvability.…”

Section: Introductionmentioning

confidence: 87%

“…A first result in this vein is attributed to Yashenko in [Pav,considerations after Corollary 3.11], where it is stated that every Tychonoff connected space is resolvable-actually, c-resolvable. In the present paper we show that the hypothesis of complete regularity in Yashenko's result may be relaxed to regularity, even if in this case we can prove only ω-resolvability.…”

Section: Introductionmentioning

confidence: 99%

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On the resolvability of locally connected spaces

Costantini¹

2004

Proc. Amer. Math. Soc.

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Abstract. We solve a problem of Padmavally about resolvability of locally connected spaces, in the case where the space under consideration is regular. IntroductionAs is well known, a space X is said to be resolvable (more generally, ν-resolvable for some cardinal ν ≥ 2) if it may be written as the union of two (respectively, ν-many) disjoint dense subspaces. The introduction of this property was essentially motivated by the discovery of crowded (i.e., without isolated points) topological spaces that fail to have it, even if this kind of space seems to be quite rare "in nature". As a matter of fact, many topological spaces with "reasonable" properties (such as, for example, locally compact spaces, or k-spaces-hence also metrizable, first-countable, sequential andČech-complete spaces) turn out to be resolvable, whenever they contain no isolated point. From this point of view, the notion of irresolvable space is to be considered, in the realm of crowded spaces, by the side of other stronger (and more "unnatural") topological properties, like maximality, submaximality, and the notions of nodec and door space (see [AC] for an overview).In this vein, it is natural to investigate if other well-known topological properties could imply resolvability. For example, the question of whether connectedness together with some suitable separation axiom implies non-submaximality (or nonmaximality, or resolvability) is implicitly raised by Hewitt in 1943, since Theorem 16 of [He] shows that for every infinite cardinal number ν there is a connected submaximal T 1 space of cardinality ν (actually, the argument given by Hewitt is needlessly complicated, because taking on any infinite set X a non-principal ultrafilter as a topology for it, makes X a connected maximal T 1 space). In [Pad, Theorem 2], Padmavally also proves that every Hausdorff submaximal space cannot be locally connected at any point, and in his reflections before such

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

On the resolvability of locally connected spaces

Costantini¹

2004

Proc. Amer. Math. Soc.

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“…For example, years ago Pytke ′ ev [31] showed that every k-space, also every space X for which the tightness t(X) satisfies t(X) < ∆(X), is maximally resolvable. More recently, denoting by ps(X) the smallest successor cardinal such that every discrete set S ⊆ X satisfies |S| < ps(X), Pavlov [29] showed that every T 1 -space such that ∆(X) > ps(X) is maximally resolvable. That theorem was strengthened in two ways in [26]: No separation hypothesis on X is required, and maximal resolvability of X is established assuming only ∆(X) ≥ ps(X).…”

Section: Remark 32mentioning

confidence: 99%

Tychonoff expansions with prescribed resolvability properties

Comfort

2010

Topology and its Applications

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A method for Tychonoff expansions using independent families is introduced. Using this method we prove that every countable Tychonoff space which admits a partition into infinitely many open-hereditarily irre-solvable dense subspaces has a Tychonoff expansion that is ω-resolvable but not strongly extraresolvable. We also show that, under Luzin's Hypothesis (2 ω 1 = 2 ω), there exists an ω-resolvable Tychonoff space of size ω 1 which is not maximally resolvable.

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“…In the particular case when µ is a successor cardinal it follows from Proposition 2.1 of [12]. Lemma 3.16: If |X| = µ is a regular cardinal and T µ (X) is dense in X then X is µ-resolvable.…”

Section: Corollary 37mentioning

confidence: 99%

Resolvability and monotone normality

2008

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A space X is said to be κ-resolvable (resp., almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp., almost disjoint over the ideal of nowhere dense subsets). X is maximally resolvable if and only if it is ∆(X)-resolvable, where ∆(X) = min{|G| : G = ∅ open}.We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost µ-resolvable, where µ = min{2 ω , ω 2 }. On the other hand, if κ is a measurable cardinal then there is a MN space X with ∆(X) = κ such that no subspace of X is ω 1 -resolvable.Any MN space of cardinality < ℵω is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = ∆(X) = ℵω such that no subspace of X is ω 2 -resolvable.

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On resolvability of topological spaces

Cited by 18 publications

References 7 publications

On the resolvability of locally connected spaces

On the resolvability of locally connected spaces

Tychonoff expansions with prescribed resolvability properties

Resolvability and monotone normality

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