Globalizing weak homotopy equivalences
\textit{A. Dold} and \textit{R. Thom} introduced and studied quasifibrations and, in particular, they proved the quasifibration theorem which asserts, roughly speaking, that a map is a quasifibration if it is locally one [Ann. Math. (2) 67, 239-281 (1958; Zbl 0091.37102)]. The author generalizes this result to weak equivalences, inspired by the development of theory of quasifibrations based on weak equivalences [\textit{J. P. May}, Lect. Notes Math. 1425, 91-101 (1990; Zbl 0713.55010). The author introduces the notions of an \(n\)-distinguished pair and of a local \(n\)-equivalence: Let \(p:X\to Y\) be a map and \(n\) a non-negative integer. For subspaces \(U,V\) of \(Y\) with \(V\subset U\) the pair \((U,V)\) is called \(n\)-distinguished with respect to \(p\) if the induced map \((p^{-1}(U),\;p^{-1}(V)) \to(U,V)\) is an \(n\)-equivalence. For an open cover \({\mathcal U}\) of \(Y\), the pair \((p,{\mathcal U})\) is called a local \(n\)-equivalence, if \({\mathcal U}\) satisfies the following two conditions, for \(U,V\in {\mathcal U}:(1)\) If \(V\subset U\), then the pair \((U,V)\) is \(n\)-distinguished with respect to \(p\), (2) to every \(x\in U\cap V\) there exists \(W\in{\mathcal U}\) such that \(x\in W\) and \(W\subset U\cap V\). The proof of the following main result is based on the proof of the Dold-Thom quasifibration theorem: Let \(p:X\to Y\) be a map and suppose that \({\mathcal U}\) is an open cover of \(Y\), such that for some non-negative integer \(n\), \((p,{\mathcal U})\) is a local \(n\)-equivalence. Then for every \(U\in{\mathcal U}\) the pair \((Y,U)\) is \(n\)-distinguished.
scientific article; zbMATH DE number 1395067