A note on a theorem of Talagrand

Moll, Salvador; Ruiz, Luis Manuel Sánchez

doi:10.1016/j.topol.2005.12.011

Cited by 2 publications

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“…So that M is a K ‐analytic subspace of

C_{p} false(Y false)

that separates the points of

Y

. According to [, Main Theorem] (which is proved only for K ‐analytic spaces) the space

C_{p} false(Y false)

is K ‐analytic. Now let us assume that

({}, x_{0})

is not a

G_{δ}

‐subset of X .…”

Section: A Characterization Of Talagrand and Gul'ko Compact Setsmentioning

confidence: 99%

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On spaces weakly ‐analytic

Ferrando

Ka̧kol

Pellicer

2017

Mathematische Nachrichten

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A subset Y of the dual closed unit ball BE* of a Banach space E is called a Rainwater set for E if every bounded sequence of E that converges pointwise on Y converges weakly in E. In this paper, topological properties of Rainwater sets for the Banach space Cbfalse(Xfalse) of the real‐valued continuous and bounded functions defined on a completely regular space X equipped with the supremum‐norm are studied. This applies to characterize the weak K‐analyticity of Cbfalse(Xfalse) in terms of certain Rainwater sets for Cbfalse(Xfalse). Particularly, we show that Cbfalse(Xfalse) is weakly K‐analytic if and only if there exists a Rainwater set Y for Cbfalse(Xfalse) such that ()Cbfalse(Xfalse),σY is both K‐analytic and angelic, where σY denotes the topology on Cbfalse(Xfalse) of the pointwise convergence on Y. For the case when X is compact, one gets classic Talagrand's theorem. As an application we show that if X is a compact space and Y is a Gδ‐dense subspace, then X is Talagrand compact, i.e., Cpfalse(Xfalse) is K‐analytic, if and only if the space (Cfalse(Xfalse),σY) is K‐analytic.

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“…So that M is a K ‐analytic subspace of

C_{p} false(Y false)

that separates the points of

Y

. According to [, Main Theorem] (which is proved only for K ‐analytic spaces) the space

C_{p} false(Y false)

is K ‐analytic. Now let us assume that

({}, x_{0})

is not a

G_{δ}

‐subset of X .…”

Section: A Characterization Of Talagrand and Gul'ko Compact Setsmentioning

confidence: 99%

“…So that is a -analytic subspace of ( ) that separates the points of . According to [22,Main Theorem] is not -analytic since ( ) is not -analytic. In any case, note that ( ( ), ) = ( ) homeomorphically due to the C-embedding in X of the pseudocompact , consequently the fact that ( ( ), )…”

Section: A Characterization Of Talagrand and Gul'ko Compact Setsmentioning

confidence: 99%