On Continuity Properties of the Modulus of Local Contractibility

Let MX be the set of all metrics compatible with a given topology on a locally contractible space X and let for each triple zs r, x, «.gMX =X= 0, `., D z. be the set of all positive d such that the open d-neighborhood of x is contractible in the open «-neighborhood of x in metric r. We prove several continuity properties of the map D : MX =X= 0, `.ª 0, `. and then, using a selection theorem for non-lower semicontinuous mappings, show that D admits a continuous singlevalued selection. Similar, but somewhat different properties are also demonstrated for the modulus Dn of local n-connectedness