A stronger limit theorem in extension theory (Q2748019)

The author, jointly with \textit{P. J. Schapiro}, had earlier proved [Pac. J. Math. 187, No. 1, 177-186 (1999; Zbl 0951.55002)] the following Limit Theorem: Suppose \({X_{i}\mid i \in \mathbb N}\) is an inverse sequence of metrizable spaces, each having the Extension Property with respect to a given simplicial complex \(K\), then their inverse limit \(X = \lim X_i\) also has the Extension Property with respect to \(K\). By an Extension Property we mean: any continuous map \(f: A \to K\) from any closed subset \(A\) of \(X_i\) can be continuously extended to \(X_i\). In the present paper, the author proves the following generalization of the above theorem. NEWLINENEWLINENEWLINETheorem: Let \({\mathbb X = (X_i, p_{i,i+1},\mathbb N)}\) be an inverse sequence of metrizable spaces, \(K\) be a CW-complex such that the inverse system \({\mathbb X}\) itself has the Extension Property with respect to \(K\). Then the inverse limit \(X = \lim X_i\) also has the Extension Property with respect to \(K\).NEWLINENEWLINENEWLINEWe say that an inverse sequence \({\mathbb X = (X_i, p_{i, i+1},\mathbb N)}\) has the Extension Property with respect to a CW-complex \(K\) if for each closed subset \(A\) of \(X_i\) and a map \(f:A \mapsto K,\) there exists a \(j \geq i\) and a map \(g: X_j \mapsto K\) such that \(g(x) = f \circ p_{j}(x)\) for every \(x \in p^{-1}_{j}(A)\). The above definition, due to A. N. Dranishnikov, is crucial in getting the foregoing generalization of the earlier theorem. It is interesting to point out that though the proof of the above generalization is quite parallel to the proof of the original theorem, yet it is remarkably different at several steps requiring careful arguments which the author has successfully done.