Comparison of Topologies on Function Spaces
نویسنده
چکیده
1. Introduction. Let A be a topological space, F be a metrizable space, and C be the collection of continuous functions on X into Y. We shall be interested in two topologies on C. The compact-open or k-topology. Given compact subset K of X and open subset W of F, denote by (K, W) the collection of functions fEC such that fiK)E W. The assembly of finite intersections of sets iK, W) forms a base for a topology on C, called the compact-open topology, or ¿-topology [l].1 Here the restriction to metrizable spaces Y is unnecessary. The topology of uniform convergence or d*-topology. If Y is metriz-able, let d be a bounded metric consistent with the topology of Y. For instance, if d' is an unbounded metric, let
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