The method of approximative extension of mappings in the theory of extensors

We develop the method of approximative extension of mappings which enables us not only to simplify the proofs of many available theorems in the theory of extensors but also to obtain a series of new results. Combined with Ancel' theory of fiberwise trivial correspondences, this method leads to considerable progress in the characterization of absolute extensors in terms of local contractivity. We prove the following assertions: Suppose that a space $X$ is represented as the union of countably many closed ANE-subspaces $X_i$ and a countably dimensional subspace $D$: 1. If each $X_i$ is a strict deformation neighborhood retract of $X$ and $X\in{\rm LC}$, then $X\in{\rm ANE}$. 2. If $X\in{\rm LEC}$, then $X\in{\rm ANE}$.