Extraresolvable spaces

A space X is called extraresolvable if there is a family D of dense subsets such that |D|>Δ(X), where Δ(X) is the dispersion character of X, and D∩D′ is nowhere dense whenever D,D′∈D and D≠D′. It is shown that if X is either a countable spaces with nowhere dense tightness or a countable (Hausdorff) weakly Fréchet–Urysohn space, then X is extraresolvable. It is not hard to see that every extraresolvable space is ω-resolvable. We prove that compact metric spaces and compact topological groups are not extraresolvable (these spaces are maximally resolvable). We also give some examples of metric extraresolvable topological Abelian groups with uncountable dispersion character, compact extraresolvable spaces with uncountable dispersion character and an example of a connected ω-bounded extraresolvable topological Abelian group.