The Lindelöf property and $\sigma $-fragmentability

B. Cascales ¹   ; I. Namioka ²

¹ Departamento de Matemáticas Universidad de Murcia 30100 Espinardo Murcia, Spain
² University of Washington Department of Mathematics Box 354350 Seattle, WA 98195-4350, U.S.A.

Fundamenta Mathematicae, Tome 180 (2003) no. 2, pp. 161-183

Résumé

In the previous paper, we, together with J. Orihuela, showed that a compact subset $X$ of the product space $[-1,1]^{D}$ is fragmented by the uniform metric if and only if $X$ is Lindelöf with respect to the topology $\gamma (D)$ of uniform convergence on countable subsets of $D$. In the present paper we generalize the previous result to the case where $X$ is $K$-analytic. Stated more precisely, a $K$-analytic subspace $X$ of $[-1,1]^{D}$ is $\sigma $-fragmented by the uniform metric if and only if $(X,\gamma (D))$ is Lindelöf, and if this is the case then $(X,\gamma (D))^{{\mathbb N}}$ is also Lindelöf. We give several applications of this theorem in areas of topology and Banach spaces. We also show by examples that the main theorem cannot be extended to the cases where $X$ is Čech-analytic and Lindelöf or countably $K$-determined.

Détail

DOI : 10.4064/fm180-2-3

Keywords: previous paper together orihuela showed compact subset product space fragmented uniform metric only lindel respect topology gamma uniform convergence countable subsets present paper generalize previous result where k analytic stated precisely k analytic subspace sigma fragmented uniform metric only gamma lindel gamma mathbb lindel several applications theorem areas topology banach spaces examples main theorem cannot extended cases where ech analytic lindel countably k determined