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Given a topological space X, let UX and ηX: X → UX denote, respectively, the Salbany compactification of X and the compactification map called the Salbany map of X. For every continuous function f: X → Y, there is a continuous function Uf: UX → UY, called the Salbany lift of f, satisfying (Uf) ◦ ηX = ηY ◦ f. If a continuous function f: X → Y has a stably compact codomain Y, then there is a Salbany extension F: UX → Y of f, not necessarily unique, such that F ◦ ηX = f. In this paper, we give a condition on a space such that its Salbany map is open. In particular, we prove that in a class of Hausdorff spaces, the spaces with open Salbany maps are precisely those that are almost discrete. We also investigate openness of the Salbany lift and a Salbany extension of a continuous function. Related to open continuous functions are initial maps as well as nearly open maps. It turns out that the Salbany map of every space is both initial and nearly open. We repeat the procedure done for openness of Salbany maps, Salbany lifts and Salbany extensions to their initiality and nearly openness. [ABSTRACT FROM AUTHOR]

In this work we study countably z-compact spaces and z-Lindelof spaces. Several new properties of them are given. It is proved that every countably z-compact space is pseuodocompact (a space on which every real valued continuous function is bounded). Spaces which are countably z-compact but not countably compact are given. It is proved that a space is countably z-compact iff every countable z-closed set is compact. Characterizations of countably z-compact and z-Lindelof spaces by multifunctions are given. [ABSTRACT FROM AUTHOR]