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 XK in the class of spaces with e-dimX⩽K together with a K-soft map onto the Hilbert cube. As a corollary we obtain Olszewskiʹs completion theorem. Also we apply the CEB Theorem to give an alternative proof of the Splitting Theorem in extension theory for noncompact spaces.