The functional characterizations of the Rothberger and Menger properties

2019

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Let USC ⋆ p (X) be the topological space of real upper semicontinuous bounded functions defined on X with the subspace topology of the product topology on X R.Φ ↑ ,Ψ ↑ are the sets of all upper sequentially dense, upper dense or pointwise dense subsets of USC ⋆ p (X), respectively. We prove several equivalent assertions to that that USC ⋆ p (X) satisfies the selection principles S 1 (Φ ↑ ,Ψ ↑ ), including a condition on the topological space X. We prove similar results for the topological space C ⋆ p (X) of continuous bounded functions. Similar results hold true for the selection principles S f in (Φ ↑ ,Ψ ↑ ).

Section: Dense Selectors Of the Space C P (X)mentioning

confidence: 67%

“…C p (X) or USC p (X) denote the set of all real continuous or upper semicontinuous functions 1 defined on the topological space X. Instead of C p (X) * or USC p (X) [14,15]). We set…”

Section: Terminology and Notationsmentioning

confidence: 99%

Selectors for dense subsets of function spaces

Bukovský

2019

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“…If n = 1, then we denote A f instead of A 1,f and A ω f instead of A ω 1,f . By Theorem 11.3 in [17], we proved the following result. Theorem 3.3.…”

Section: The Projectively Rothberger Propertymentioning

confidence: 80%

“…The remaining implications are proved in the same way as in the proof of Theorem 11.3 in [17] by replacing n-dense (dense) subsets of C p (X) with countable n-dense (dense) subsets of C p (X).…”

Section: The Projectively Rothberger Propertymentioning

confidence: 94%

Projective versions of the properties in the Scheepers Diagram

2020

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Let P be a topological property. A.V. Arhangel'skii calls X projectively P if every second countable continuous image of X is P. Lj.D.R. Kočinac characterized the classical covering properties of Menger, Rothberger, Hurewicz and Gerlits-Nagy in term of continuous images in R ω . In this paper we have gave the functional characterizations of all projective versions of the selection properties in the Scheepers Diagram.Theorem 14.2. For a s-space X with iw(X) = ω, the following statements are equivalent:

“…For a topological space X we denote: Different ∆-covers (k-covers, ω-covers, k F -covers, c F -covers,...) exposed many dualities in hyperspace topologies such as co-compact topology F + , cofinite topology Z + , Pixley-Roy topology, Fell topology and Vietoris topology. They also play important roles in selection principles ( [1,[4][5][6][7][8][9][12][13][14][15][16]).…”

Section: Introductionmentioning

confidence: 99%

Selectors for sequences of subsets of hyperspaces

2020

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For a Hausdorff space X we denote by 2 X the family of all closed subsets of X. In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. We investigate selectors for sequence of subsets of the space 2 X with the Z +topology and the upper Fell topology (F + -topology). Also we consider the selection properties of the bitopological space (2 X , F + , Z + ).