The Lindelöf property and $\sigma $-fragmentability

Volume 180 / 2003

B. Cascales, I. Namioka Fundamenta Mathematicae 180 (2003), 161-183 MSC: Primary 54C35, 46E15; Secondary 46A50. DOI: 10.4064/fm180-2-3


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.


  • B. CascalesDepartamento de Matemáticas
    Universidad de Murcia
    30100 Espinardo
    Murcia, Spain
  • I. NamiokaUniversity of Washington
    Department of Mathematics
    Box 354350
    Seattle, WA 98195-4350, U.S.A.

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