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Strong uniform continuity. (English) Zbl 1161.54003
Strong local continuity is a relative concept: if $f:X\to Y$ is a continuous map between metric spaces then not only is $f\upharpoonright K$ uniformly continuous whenever $K$ is compact: the $\delta>0$ that corresponds to the given $\epsilon>0$ satisfies the implication “if $d(x,y)<\delta$ then $d(f(x),f(y))<\epsilon$” 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 $\Cal{B}^f=\{B:f\upharpoonright B$ is uniformly continuous$\}$ and $\Cal{B}_f=\{B:f$ 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 $\Cal{B}$ the authors study the topology of strong uniform convergence on members of $\Cal{B}$ (derived from a uniformity wherein closeness of functions is required on neighbourhoods of members of $\Cal{B}$).

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