Filomat 2010 Volume 24, Issue 4, Pages: 63-75
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(Strong) weak exhaustiveness and (strong uniform) continuity
Caserta Agata, Di Maio Giuseppe, Holá L'ubica
In this paper we continue, in the realm of metric spaces, the study of exhaustiveness and weak exhaustiveness at a point of a net of functions initiated by Gregoriades and Papanastassiou in 2008. We prove that exhaustiveness at every point of a net of pointwise convergent functions is equivalent to uniform convergence on compacta. We extend exhaustiveness-type properties to subsets. First, we introduce the notion of strong exhaustiveness at a subset B for sequences of functions and prove its equivalence with strong exhaustiveness at P0 (B) of the sequence of the direct image maps, where the hypersets are equipped with the Hausdorff metric. Furthermore, we show that the notion of strong-weak exhaustiveness at a subset is the proper tool to investigate when the limit of a pointwise convergent sequence of functions fulfills the strong uniform continuity property, a new pregnant form of uniform continuity discovered by Beer and Levi in 2009.
Keywords: Pointwise convergence, quasi-uniform convergence on compacta, bornology, uniform continuity, strong uniform convergence, exhaustiveness, weak exhaustivenesss at points, at subsets
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