On absolute extensors modulo a complex
Let \(L\) be a CW complex. The notation \(\operatorname {ed}(X)\leq [L]\) reads that the extensional dimension of \(X\) does not exceed \([L]\) and it means that every continuous map \(f:A\to L\), defined on a closed subset \(A\subset X\), has a continuous extension. We note that the condition \(\operatorname {ed}(X)\leq [S^n]\) means exactly that \(\dim X\leq n\). By this analogy a space \(X\) with \(\operatorname {ed}(X)\leq [L]\) is called at most \([L]\)-dimensional. Let \({\mathcal L}\) denote the class of all at most \([L]\)-dimensional spaces. Let \(A(N)E([L])\) be the class of absolute (neighborhood) extensors for the class \({\mathcal L}\). The main result of the paper is the following Theorem. Let \(L\) be a noncontractible connected CW complex. Then every \([L]\)-dimensional \(A(N)E([L])\) compact space \(X\) is metrizable. This is a generalization to the extension dimension of a theorem from ordinary dimension theory [the reviewer, Russ. Math. Surv. 39, No. 5, 63-111 (1984); translation from Usp. Mat. Nauk 39, No. 5(239), 55-95 (1984; Zbl 0572.54012)] which states that \(n\)-dimensional \(A(N)E(n)\) compact space is metrizable for \(n>0\).
scientific article; zbMATH DE number 1307972
scientific article