Function Spaces and Strong Variants of Continuity

Abstract

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.

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