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Strong uniform continuity. (English) Zbl 1161.54003

Strong local continuity is a relative concept: if f:XY is a continuous map between metric spaces then not only is fK uniformly continuous whenever K is compact: the δ>0 that corresponds to the given ϵ>0 satisfies the implication “if d(x,y)<δ then d(f(x),f(y))<ϵ” even when just one of x and y belongs to K. This state of affairs is abbreviated as: f is strongly uniformly continuous on K. The authors study this concept in some depth. They compare the families  f ={B:fB is uniformly continuous} and f ={B:f is strongly uniformly continuous on B}; the latter is an ideal (and a bornology if f is continuous), the former need not be.

In the second part of the paper the attention shifts to function space topologies; for a bornology  the authors study the topology of strong uniform convergence on members of  (derived from a uniformity wherein closeness of functions is required on neighbourhoods of members of ).

54C05Continuous maps
54C10Special maps on topological spaces
54C35Function spaces (general topology)
54E15Uniform structures and generalizations
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