Arzelà’s theorem and strong uniform convergence on bornologies. (English) Zbl 1202.54004

Caserta, Agata; Di Maio, Giuseppe; Holá, Ľubica

J. Math. Anal. Appl. 371, No. 1, 384-392 (2010).

The main result of this nice paper (Theorem 2.9) gives a direct proof that three kinds of convergence (Arzelà’s convergence on compacta, Alexandroff’s convergence and strong uniform convergence on finite sets, introduced recently by G. Beer and S. Levi [J. Math. Anal. Appl. 350, 568–589 (2009; Zbl 1161.54003)]) of nets of continuous functions between two metric spaces coincide, and that each of these conditions is equivalent to continuity of the limit function. Several properties ((sub)metrizability, the \(G_\delta\)-diagonal property, countable pseudocharacter, separability, second countability, the Fréchet-Urysohn property) of the topology of strong uniform convergence on a bornology are also established.

Reviewer: Ljubiša D. Kočinac (Niš)

MSC:

54A20	Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54C05	Continuous maps
54C35	Function spaces in general topology

Keywords:

Arzelà’s quasi-uniform convergence; pointwise and uniform convergence; strong continuity; strong uniform convergence; bornology

Citations:

Zbl 1161.54003

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