Fragmentability and compactness in C(K)-spaces

B. Cascales; G. Manjabacas; G. Vera

Studia Mathematica (1998)

  • Volume: 131, Issue: 1, page 73-87
  • ISSN: 0039-3223

Abstract

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Let K be a compact Hausdorff space, C p ( K ) the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and t p ( D ) the topology in C(K) of pointwise convergence on D. It is proved that when C p ( K ) is Lindelöf the t p ( D ) -compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and C p ( K ) is Lindelöf, then K is metrizable if, and only if, there is a countable and dense subset D ⊂ K such that ( C ( K ) , t p ( D ) ) is analytic. We also show that if K is a separable Rosenthal compact space, then K is metrizable if, and only if, C p ( K ) is Lindelöf. We complete our study by showing that if K does not contain a copy of βℕ, then convex t p ( D ) -compact subsets of C(K) have the weak Radon-Nikodym property.

How to cite

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Cascales, B., Manjabacas, G., and Vera, G.. "Fragmentability and compactness in C(K)-spaces." Studia Mathematica 131.1 (1998): 73-87. <http://eudml.org/doc/216564>.

@article{Cascales1998,
abstract = {Let K be a compact Hausdorff space, $C_p(K)$ the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and $t_p(D)$ the topology in C(K) of pointwise convergence on D. It is proved that when $C_p(K)$ is Lindelöf the $t_p(D)$-compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and $C_p(K)$ is Lindelöf, then K is metrizable if, and only if, there is a countable and dense subset D ⊂ K such that $(C(K),t_p(D))$ is analytic. We also show that if K is a separable Rosenthal compact space, then K is metrizable if, and only if, $C_p(K)$ is Lindelöf. We complete our study by showing that if K does not contain a copy of βℕ, then convex $t_p(D)$-compact subsets of C(K) have the weak Radon-Nikodym property.},
author = {Cascales, B., Manjabacas, G., Vera, G.},
journal = {Studia Mathematica},
keywords = {pointwise compactness; Radon-Nikodym compact spaces; fragmentability; pointwise convergence topology; Lindelöf; Radon-Nikodým compact space},
language = {eng},
number = {1},
pages = {73-87},
title = {Fragmentability and compactness in C(K)-spaces},
url = {http://eudml.org/doc/216564},
volume = {131},
year = {1998},
}

TY - JOUR
AU - Cascales, B.
AU - Manjabacas, G.
AU - Vera, G.
TI - Fragmentability and compactness in C(K)-spaces
JO - Studia Mathematica
PY - 1998
VL - 131
IS - 1
SP - 73
EP - 87
AB - Let K be a compact Hausdorff space, $C_p(K)$ the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and $t_p(D)$ the topology in C(K) of pointwise convergence on D. It is proved that when $C_p(K)$ is Lindelöf the $t_p(D)$-compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and $C_p(K)$ is Lindelöf, then K is metrizable if, and only if, there is a countable and dense subset D ⊂ K such that $(C(K),t_p(D))$ is analytic. We also show that if K is a separable Rosenthal compact space, then K is metrizable if, and only if, $C_p(K)$ is Lindelöf. We complete our study by showing that if K does not contain a copy of βℕ, then convex $t_p(D)$-compact subsets of C(K) have the weak Radon-Nikodym property.
LA - eng
KW - pointwise compactness; Radon-Nikodym compact spaces; fragmentability; pointwise convergence topology; Lindelöf; Radon-Nikodým compact space
UR - http://eudml.org/doc/216564
ER -

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