Extension property and \(\text{ANR}\)-systems (Q1581958)

The author studies some variants of the Kuratowski notion of extendability \(\tau \), where \(Y\tau X\) means that every map \(g:A\rightarrow X\) defined on a closed subset \(A\) of a space \(Y\) can be extended to a map \(G:Y\rightarrow X\). He defines, in particular, the extension property \(Y\tau {\mathcal S}\) where \({\mathcal S}\) is an ANR-system and proves a sum theorem for the class of spaces \(Y\) such that \(Y\tau {\mathcal S}\). He also proves, among other results, that if \(Y\tau {\mathcal S}\) then \(Y\tau SP^{n}({\mathcal S})\), where \(SP^{n}\) denotes the symmetric power functor. This generalizes a theorem of Dranishnikov.