ON CONTINUITY PROPERTIES OF THE MODULUS OF LOCAL CONTRACTIBILITY

Let M-X be the set of all metrics compatible with a given topology on a locally contractible space X and let for each triple z = (rho, x, ep silon) is an element of M-X x X x (0, infinity), Delta(z) be the set o f all positive delta such that the open delta-neighborhood of x is con tractible in the open epsilon-neighborhood of x in metric rho. We prov e several continuity properties of the map Delta : M-X x X x (0, infin ity) --> (0, infinity) and then, using a selection theorem for non-low er semicontinuous mappings, show that Delta admits a continuous single valued selection. Similar, but somewhat different properties are also demonstrated for the modulus Delta(n) of local n-connectedness. (C) 19 97 Academic Press.