A notion of true dimension
theory is defined to which is assigned a dimension function D. We consider those D
which have an enhanced Bockstein basis; these include D =dim and D =dimG for
any abelian group G. We prove that for each countable polyhedron K, the set of
compacta X ∈ 2Q with K ∈AE({X}) is a Gδ-subspace. We apply this fact to show
that the hyperspace of the Hilbert cube Q consisting of compacta (or continua) X
with D(X) ≤ n is a Gδ-subspace. Let D≥n (resp., D≥n∩C(Q)) denote the space of
compacta X (resp., continua) with D(X) ≥ n. We prove that {D≥n}n=1∞ and
{D≥n∩ C(Q)}n=2∞ are absorbing sequences for σ-compact spaces. This
yields that each D≥n and D≥n+1∩ C(Q)(n ≥ 1) is homeomorphic to the
pseudoboundary B of Q; their respective complements are homeomorphic to the
pseudointerior of Q; and the intersections ⋂nD≥n, ⋂nD≥n∩ C(Q) are
homeomorphic to B∞, the absorbing set for the class of Fσδ- sets. Results for
the hyperspaces of compacta X for which D(X) ≥ n uniformly are also
obtained.
