A Criterion for Local Resolvability of a Space and the w-Problem

The notion of the resolvability of a topological space was introduced by E. Hewitt (8). Recently it was understood that this no- tion is also important in the study of !-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 posi- tive continuous real-valued function has an !-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 !-primitive.