Extensions of unbounded topological spaces

A method of compactification of locally compact spaces, based on the concept of essential semilattice homomorphism, has been recently proposed. In this paper we present a natural generalization of that construction. We say that a topological space X is locally bounded with respect to a boundedness F, that is a family of subsets of X closed under finite unions and subsets, if every point of X has a bounded neighborhood. Clearly, if F is the family of relatively compact subsets of X (relatively Lindelöf subsets of a regular space X), local boundedness with respect to F is equivalent to local compactness (resp. local Lindelöfness) of X. We construct dense extensions of unbounded spaces, called B-extensions. By adding some requirements conditions, we obtain extensions that satisfy some separation and covering properties.This method can be applied, for instance, to construct Lindelöf extensions of locally Lindelöf spaces.