Function Spaces and Strong Variants of Continuity
Keywords:Strongly continuous function, Perfectly continuous function, cl-supercontinuous function, Sum connected spaces, k-space, Topology of point wise convergence, Topology of uniform convergence on compacta, Compact open topology, Equicontinuity, Even continuit
It is shown that if domain is a sum connected space and range is a T0-space, then the notions of strong continuity, perfect continuity and cl-supercontinuity coincide. Further, it is proved that if X is a sum connected space and Y is Hausdorff, then the set of all strongly continuous (perfectly continuous, cl-supercontinuous) functions is closed in Y X in the topology of pointwise convergence. The results obtained in the process strengthen and extend certain results of Levine and Naimpally.
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