The Countable Extension Basis Theorem and its applications

We prove the Countable Extension Basis (CEB) Theorem for noncompact metric spaces. As an application of the CEB Theorem we give alternative proofs of some classical results in the area, In particular, we construct a universal complete metric space X-K in the class of spaces with e-dim X less than or equal to K together with a K-soft map onto the Hilbert cube. As a corollary we obtain Olszewski's completion theorem, Also we apply the CEB Theorem to give an alternative proof of the Splitting Theorem in extension theory for noncompact spaces. (C) 2001 Elsevier Science B.V. All rights reserved.