A criterion for local resolvability of a space and the ω-problem

The notion of the resolvability of a topological space was introduced by E. Hewitt [8]. Recently it was understood that this notion is also important in the study of ω)-primitives, especially in the case of nonmetrizable spaces. In the present paper a criterion for the resolvability of a topological space at a point ("local resolvability") is given. This criterion, stated in terms of oscillation and quasicontinuity, permits to conclude, for instance, that on irresolvable spaces no positive continuous real-valued function has an ω-primitive. The result is strenghtened in the case of SI-spaces. It is also shown that every non- negative upper semicontinuous function on a resolvable Baire space has an ω)-primitive.