THE HYPERSPACES OF INFINITE-DIMENSIONAL COMPACTA FOR COVERING AND COHOMOLOGICAL DIMENSION ARE HOMEOMORPHIC

A notion of true dimension theory is defined to which is assigned a di mension function D. We consider those D which have an enhanced Bockste in basis; these include D = dim and D = dim(G), for any abelian group G. We prove that for each countable polyhedron K, the set of compacta X is-an-element-of 2Q with K is-an-element-of AE({X}) is a G(delta)-su bspace. We apply this fact to show that the hyperspace of the Hilbert cube Q consisting of compacta (or continua) X with D(X) less-than-or-e qual-to n is a G(delta)-subspace. Let D(greater-than-or-equal-to n) (r esp., D(greater-than-or-equal-to n) and C(Q)) denote the space of comp acta X (resp., continua) with D(X) greater-than-or-equal-to n. We prov e that {D(greater-than-or-equal-to n)n=1 infinity and {D(greater-than- or-equal-to n) and C(Q)}n=2 infinity are absorbing sequences for sigma -compact spaces. This yields that each D(greater-than-or-equal-to n) a nd D(greater-than-or-equal-to n+1) and C(Q) (n greater-than-or-equal-t o 1) is homeomorphic to the pseudoboundary B of Q; their respective co mplements are homeomorphic to the pseudointerior of Q; and the interse ctions and n D(greater-than-or-equal-to n), and n D(greater-than-or-eq ual-to n) and C(Q) are homeomorphic to B(infinity), the absorbing set for the class of F(sigmadelta)-sets. Results for the hyperspaces of co mpacta X for which D(X) greater-than-or-equal-to n uniformly are also obtained.