Almost-homeomorphisms and aumosttopological properties

A function is said to be an almosthomeomorphism if it is a bijective almost continuous function (see [25]) with an almost continuous inverse. We characterize such functions in several ways and obtain the relationship between almost-homeomorphisms and semi-homeomorphisms (see [8]). We study those properties which are preserved under this class of functions -the almost  topological properties - and characterize them as the semi-regular properties (see [3]). We also introduce the concept of an almost topological class and study the relationship between this clases and the topological, semi-topological, and p-topological classes.