Preservation of generalized continuity. (English) Zbl 0723.54011

Di Maio, Giuseppe; Naimpally, Somashekhar A.

J. Nat. Phys. Sci. 3, No. 1-2, 90-104 (1989).

Various types of convergence of sequences of p-quasi-continuous and p- nearly continuous functions between preproximity spaces are investigated. A basic preproximity on a set X is a symmetric binary relation \(\delta\) on \({\mathcal P}(X)\) extending the incidence relation and such that \(\delta\)- near sets are nonempty, \(A\delta\) B, \(B\subset C\) implies \(A\delta\) C. Three modifications of a preproximity are also considered. Except of the relationships between preproximities and a pretopology (which is any family of subsets of X containing \(\emptyset\), X and closed under arbitrary unions) the authors compare types of a convergence and treat the preservation of mentioned continuity types with respect to considered convergences. Some examples for topological spaces, especially for subspaces of R are also included.

Reviewer: J.Chvalina (Brno)

MSC:

54C08	Weak and generalized continuity
54C35	Function spaces in general topology
54E05	Proximity structures and generalizations
54E15	Uniform structures and generalizations
54C05	Continuous maps

Keywords:

uniform convergence; Arzela convergence; Dini convergence; quasi- continuity; near-continuity; preproximities