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1998 | 131 | 1 | 73-87
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Fragmentability and compactness in C(K)-spaces

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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.
  • Departamento de Matemticas, Facultad de Matemticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
  • Departamento de Matemáticas, Escuela Universitaria Politécnica, Avenida de España s/n, Campus Universitario, 02071 Albacete, Spain
  • Departamento de Matemticas, Facultad de Matemticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
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