Convergent free sequences in compact spaces

Juhász, István; Szentmiklóssy, Zoltán

doi:10.1090/s0002-9939-1992-1137223-8

Cited by 28 publications

(21 citation statements)

References 3 publications

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“…We will see in Theorem 2.8 that a similar application of elementary sequences will imply that a csD space will have a property stronger than countable tightness. This approach was inspired by the Juhász-Szentmiklóssy proof from [6] where it is shown that if a compact space does not have countable tightness, then it will contain a converging (free) ω 1 -sequence (also making essential use of Sapirovskiȋ's result). A csD space can not contain a (co-countably) converging ω 1sequence.…”

Section: 2mentioning

confidence: 99%

“…There are a number of very interesting results known for csD spaces and we recommend [4] as an excellent reference. In particular, it is known that csD spaces have countable tightness ( [6]) and that it follows from CH that csD spaces are metrizable ( [5,6]). One of our main results is that ccc subspaces of a csD space have countable π-weight.…”

Section: Introductionmentioning

confidence: 99%

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Elementary chains and compact spaces with a small diagonal

Dow

Hart

2012

Indagationes Mathematicae

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It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of submodels. We prove that ccc subspaces of such spaces have countable \pi-weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character

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Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Elementary chains and compact spaces with a small diagonal

Dow

Hart

2012

Indagationes Mathematicae

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“…Since H. Zhou had shown in [56] it to be consistent with ZFC that every compact Hausdorff space with a small diagonal is metrizable, Arkhangel'skii and Tkačhuk were able to obtain the relative consistency of the assertion that a compact Hausdorff space K is metrizable if C p (K ) is Shanin. I. Juhász and Z. Szentmiklóssy showed in [26] that every compact space with a small diagonal is metrizable under the Continuum Hypothesis (CH). It follows that, under CH, [7]; it is still an open problem (see [7, Problem 68]) whether this result holds without any additional set-theoretic assumptions.…”

Section: Chain Conditions For Banach Spacesmentioning

confidence: 99%

Chain conditions and weak topologies

Dow

Junnila

Pelant

2009

Topology and its Applications

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We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map E → c 0 (ω 1 ) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C (K ), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space C p (K ) may be pcc even when C (K ) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example: -There exists a non-metrizable scattered compact Hausdorff space K with C (K ) weakly pcc. Lemma. Let Γ be a set, and let S be a subset of c 0 (Γ ) such that S is pcc in the relative weak topology. Then S is separable.Proof. It suffices to show that the set Γ = {γ ∈ Γ : y γ = 0 for some y ∈ S} is countable. Assume that Γ is uncountable.Then there exists r > 0 such that the set Γ = {γ ∈ Γ : |y γ | > r for some y ∈ S} is uncountable. For every γ ∈ Γ , let G γ = {y ∈ S: |y γ | > r}, and note that G γ is a non-empty relatively weakly open subset of S. For every y ∈ S, the set {γ ∈ Γ : y ∈ G γ } is finite. It follows that {G γ : γ ∈ Γ } is an uncountable point-finite family of relatively weakly open subsets of S, and this is a contradiction. 2 Proposition. The following are equivalent for a Banach space E:A. E is weakly pcc.B. For any Γ , every weak-to-weak continuous mapping E → c 0 (Γ ) has separable range. C. Every weak-to-weak continuous mapping E → c 0 (ω 1 ) has separable range.Proof. The implication A ⇒ B follows by Lemmas 1.2(a) and 1.4. C ⇒ A: Assume that E is not weakly pcc. Then there exists a point-finite family {U α : α ∈ ω 1 } consisting of distinct weakly open subsets of E. For every α ∈ ω 1 , there exists a non-constant, weakly continuous function f α : E → I such that

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“…Juhász and Szentmiklóssy established in [12] that if the diagonal of a compact space X is small then X has countable tightness so CH is sufficient for metrizability of a compact space with a small diagonal. Since then, quite a few results were obtained about spaces with a small diagonal (see e.g., [9]).…”

Section: Theorem the Space C P (X) Is ω-Embedded In A σ -Compact Framentioning

confidence: 99%

Embeddings and frames of function spaces

Tkachuk

2009

Journal of Mathematical Analysis and Applications

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For every Tychonoff space X we denote by C p (X) the set of all continuous real-valued functions on X with the pointwise convergence topology, i.e., the topology of subspaceembeds in a σ -compact space of countable tightness then X is countable. This shows that it is natural to study when C p (X) has a frame of countable tightness with some compactness-like property. We prove, among other things, that if X is compact and the space C p (X) has a Lindelöf frame of countable tightness then t( X) ω. We give some generalizations of this result for the case of frames as well as for embeddings of C p (X) in arbitrary spaces.© 2009 Elsevier Inc. All rights reserved. IntroductionThe space C p (X) seldom has a compactness-like property; for example, C p (X) is a countable union of its countably compact subspaces if and only if X is finite [14]. However, for any pseudocompact space X it is possible to find a σ -compact frame for C p (X), i.e., there exists a σ -compact set P such that C p (X) ⊂ P ⊂ R X . It is a result of Uspenskij that for any Lindelöf Σ -space X there exists a Lindelöf Σ -space P which is a frame for C p (X). These results show the importance of studying the spaces X for which nice frames for C p (X) exist. Arhangel'skii [2] and Okunev [13] were the first ones to study and systematize the relevant classes of spaces. In [13], Okunev gave characterizations, in terms of topology of X , for existence of σ -compact frames, Lindelöf Σ -frames and many other kinds of frames for C p (X).Any result about nice frames of C p (X) actually says something about embeddings of C p (X) and sometimes, a possibility to embed C p (X) in a space with nice properties implies strong restrictions on X . We show that if C p (X) embeds in a σ -compact space of countable tightness then X is countable. This illustrates the importance of studying frames of countable tightness for C p (X).A theorem of Asanov [5] states that if C p (X) is Lindelöf then tightness of X is countable. It would be natural to try to obtain the same conclusion for the spaces X with a Lindelöf frame of countable tightness for C p (X). We prove that if X is a compact space and C p (X) has a Lindelöf frame of countable tightness then t( X) ω; an example is constructed which shows that compactness of X cannot be omitted in this result.We also establish that if the space C p (X) can be ω-embedded in a σ -compact space Z (this means that every point of Z is in the closure of a countable subset of C p (X)) then X has a small diagonal. Therefore, if X is a Lindelöf Σ -space and C p (X) can be ω-embedded in a σ -compact space then, under the Continuum Hypothesis, the space X has a countable network.

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Convergent free sequences in compact spaces

Cited by 28 publications

References 3 publications

Elementary chains and compact spaces with a small diagonal

Elementary chains and compact spaces with a small diagonal

Chain conditions and weak topologies

Embeddings and frames of function spaces

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