Perfect maps on convergence spaces

Abstract

The concept of the perfect map on a convergence space (X, q), whereqis a convergence function, is introduced and investigated. Such maps are not assumed to be either continuous or surjective. Some nontrivial examples of well known mappings between topological spaces, nontopological pretopological spaces and nonpseudotopological convergence spaces are shown to be perfect in this new sense. Among the numerous results obtained is a covering property for perfectness and the result that such maps are closed, compact, and for surjections almost-compact. Sufficient conditions are given for a compact (respectively almost-compact) map to be perfect. In the final section, a major result shows that iff: (X, q) → (Y, p) is perfect andg: (X, q) → (Z, s) is weakly-continuous into HausdorffZ, then (f, g): (X, q) → (Y×Z,p×s) is perfect. This result is given numerous applications.