Pseudocompactness of hyperspaces

Hrušák, Michael; Hernández-Hernández, Fernando; Martínez-Ruiz, Iván

doi:10.1016/j.topol.2007.07.001

Cited by 9 publications

(11 citation statements)

References 5 publications

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“…Notice that since A α+1 ∈ p for each α, p − lim g α exists, therefore p − lim F exists. By combining the above lemma and Theorem 3.2, we get Theorem 3.2 of [7], which states that under p = c, CL(Ψ(A)) is pseudocompact for every mad family A.…”

Section: Proof We Apply Theorem 22 By Lettingmentioning

confidence: 86%

“…Therefore, CL(X) is not pseudocompact. Theorem 3.2 may also be used to give a simpler proof of Theorem 3.2 of [7] as we will see, that states that under p = c, CL(Ψ(A)) is pseudocompact for every mad family A. S. Shelah and M. Malliaris have recently proved that p = t ([10]), however, we avoid the complicated model theoretic proof of p = t. Lemma 3.7. For every mad family A, ω t is relatively countably compact in Ψ(A) t , thus, Ψ(A) t is (t, ω * )−pseudocompact.…”

Section: Proof We Apply Theorem 22 By Lettingmentioning

confidence: 99%

“…In [5], J. Ginsburg has raised the question whether there is a relationship between the pseudocompactness of X ω and CL(X). He also proved that if every power of a space X is countably compact then so is CL(X), and that if CL(X) is countably compact (pseudocompact) then so is every finite power of X. Michael Hrušák, Fernando Hernández-Hernández and Iván Martínez-Ruiz proved in [7] that under h < c, there exists a mad family A such that CL(Ψ(A)) is not pseudocompact, and that under p = c, for every mad family A, CL(Ψ(A)) is pseudocompact. They also constructed, in ZFC, a subspace of βω whose hyperspace is not compact but its ω-th power is pseudocompact, showing that under the axioms of ZFC, it is not true that the pseudocompactness of X ω implies the pseudocompactness of CL(X).…”

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

Small Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $βω$

Ortiz-Castillo,

Rodrigues,

Tomita

2017

Preprint

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We study the relations between a generalization of pseudocompactness, named (κ, M )-pseudocompactness, the countably compactness of subspaces of βω and the pseudocompactness of their hyperspaces. We show, by assuming the existence of c-many selective ultrafilters, that there exists a subspace of βω that is (κ, ω * )-pseudocompact for all κ < c, but CL(X) isn't pseudocompact. We prove in ZFC that if ω ⊆ X ⊆ βω is such that X is (c, ω * )-pseudocompact, then CL(X) is pseudocompact, and we further explore this relation by replacing c for some small cardinals. We provide an example of a subspace of βω for which all powers below h are countably compact whose hyperspace is not pseudocompact, we show that if ω ⊆ X, the pseudocompactness of CL(X) implies that X is (κ, ω * )-pseudocompact for all κ < h, and provide an example of such an X that is not (b, ω * )-pseudocompact.

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Section: Proof We Apply Theorem 22 By Lettingmentioning

confidence: 86%

Section: Proof We Apply Theorem 22 By Lettingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

Small Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $βω$

Ortiz-Castillo,

Rodrigues,

Tomita

2017

Preprint

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“…In [4,5,12,13,18,20] some spaces which may be embedded as G δ -subsets of pseudocompact spaces were studied. In [19] the problem of pseudocompactness of the exponent was approached. In the present article we continue to investigate the G δ -subspaces of pseudocompact spaces.…”

Section: Introductionmentioning

confidence: 99%

On paracompactGδ-subspaces of pseudocompact spaces

Choban¹

2015

Topology and its Applications

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“…To complete the analogy, by propositions 0.4.23 and 0.4.24, we would need a theorem such as "if X is Tychonoff and that X ω is pseudocompact, then exp(X) is pseudocompact". It turns out that this result is false, as firstly proved in [43], but J. Ginsburg did not know about this at that time. We will discuss more about that in the next sections.…”

Section: Ultrafilter Versionmentioning

confidence: 96%

Weakenings of compactness and normality on Isbell-Mrówka spaces, Hyperspaces of Vietoris and Abelian groups

Rodrigues¹

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Nós fornecemos um exemplo de espaço topológico Tychonoff, almost-normal não normal e exploramos almost-normalidade restrita aos espaços de Isbell-Mrówka. Seguindo essa linha de estudo, estudamos almost disjoint families fortemente ℵ 0 -separadas comparando elas ao que se sabe sobre almost disjoint families normais e pseudonormais. Definimos uma nova família de conjuntos especiais de números reais relacionadas a esses problemas que chamamos de weak λ-sets. Esse estudo explora algumas questões de Paul Szeptycki e Sergio García-Balan.Nós exploramos as perguntas de John Ginsburg sobre pseudocompacidade e compacidade enumerável de hiperespaços de Vietoris. Em particular, obtivemos um exemplo de um subespaço de βω contendo ω cujas todas potências menores do que a característica cardinal h são enumeravelmente compactas, mas cujo hiperespaço de Vietoris não é pseudocompacto. Também exploramos essas perguntas restritas a espaços de Isbell-Mrówka, provando que a existência de uma MAD family cujo hiperespaço de Vietoris de seu espaço de Isbell-Mrówka não é pseudocompacto é equivalente ao número de Baire de ω * ser menor ou igual à c. Também obtivemos um exemplo consistente de um espaço de Isbell-Mrówka deste tipo de cardinalidade ω 2 < c.Finalmente, utilizamos forcing para obter uma classificação para grupos Abelianos de não torção de cardinalidade ≤ 2 c que admitem uma topologia enumeravelmente compacta Hausdorff contendo sequências convergentes, parcialmente respondendo uma questão de Dikranjan and Shakhmatov.

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Pseudocompactness of hyperspaces

Cited by 9 publications

References 5 publications

Small Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $βω$

Small Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $βω$

On paracompactGδ-subspaces of pseudocompact spaces

Weakenings of compactness and normality on Isbell-Mrówka spaces, Hyperspaces of Vietoris and Abelian groups

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