Vol. 51, No. 2, 1974

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Separate continuity and joint continuity

Isaac Namioka

Vol. 51 (1974), No. 2, 515–531

The main theorem is somewhat stronger than the following statement: Let X be either a locally compact Hausdorff space of a complete metric space, let Y be a compact Hausdorff space and let Z be a metric space. If a map f : X × Y Z is separately continuous, then there is a dense Gδ-set A in X such that f is jointly continuous at each point of A × Y . This theorem has consequences such as Ellis’ theorem on separately continuous actions of locally compact groups on locally compact spaces and the existence of denting points on weakly compact convex subsets of locally convex metrizable linear topological spaces.

Mathematical Subject Classification 2000
Primary: 54C05
Secondary: 46A99
Received: 17 October 1973
Published: 1 April 1974
Isaac Namioka