Extensions defined using bornologies

Many extensions of a space X such that the remainder Y is closed can be constructed as B-extensions, that is, by defining a topology on the disjoint union of X and Y, provided there exists a map, satisfying suitable conditions, from a basis of Y into the family of the subsets of X whch are unbounded with respect to a given bornology in X. In this paper we give a first example of a (nonregular) extension with closed remainder which cannot be obtained as B-extension. We also answer some open questions about separation properties and metrizability of B-extensions.