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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 $f↾K$ uniformly continuous whenever $K$ is compact: the $\delta >0$ that corresponds to the given $ϵ>0$ satisfies the implication “if $d\left(x,y\right)<\delta$ then $d\left(f\left(x\right),f\left(y\right)\right)<ϵ$” 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 ${ℬ}^{f}=\left\{B:f↾B$ is uniformly continuous$\right\}$ and ${ℬ}_{f}=\left\{B:f$ is strongly uniformly continuous on $B\right\}$; 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 $ℬ$ the authors study the topology of strong uniform convergence on members of $ℬ$ (derived from a uniformity wherein closeness of functions is required on neighbourhoods of members of $ℬ$).

##### MSC:
 54C05 Continuous maps 54C10 Special maps on topological spaces 54C35 Function spaces (general topology) 54E15 Uniform structures and generalizations
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