The Lindelöf property and $\sigma $-fragmentability
Volume 180 / 2003
Abstract
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.
