Strong uniform continuity. (English) Zbl 1161.54003

Beer, Gerald; Levi, Sandro

J. Math. Anal. Appl. 350, No. 2, 568-589 (2009).

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 \(\mathcal{B}^f=\{B:f\upharpoonright B\) is uniformly continuous\(\}\) and \(\mathcal{B}_f=\{B:f\) 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 \(\mathcal{B}\) the authors study the topology of strong uniform convergence on members of \(\mathcal{B}\) (derived from a uniformity wherein closeness of functions is required on neighbourhoods of members of \(\mathcal{B}\)).

Reviewer: K. P. Hart (Delft)

MSC:

54C05	Continuous maps
54C10	Special maps on topological spaces (open, closed, perfect, etc.)
54C35	Function spaces in general topology
54E15	Uniform structures and generalizations

Keywords:

uniform continuity; strong uniform continuity; bornology; oscillation

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