A remark on proper partitions of unity (Q452108)

Recall that a covering of a space is numerable if it admits a refinement by a partition of unity. For instance, any open covering of a paracompact space is numerable.NEWLINENEWLINEIn the paper under review, the autor first extends this notion to the proper setting by defining a covering \({\mathcal U}=\{U_i\}_{i\in I}\) to be proper numerable if \({\mathcal U}^\infty=\bigl\{U_i\cup\{\infty\}\bigr\}_{i\in I}\) is a numerable covering of the Alexandroff compactification.NEWLINENEWLINEThen, it is proved that, in a finite dimensional locally finite CW-complex, any open covering containing a cocompact set is proper numerable.NEWLINENEWLINEFinally, an interesting version of a classical result of Dold on fibrewise homotopy equivalences is proved in the proper setting. Namely, let \(p: X\rightarrow B\leftarrow Y: q\) be maps and let \(f: X\to Y\) be a map over \(B\), i.e., \(qf=p\). If \(\{B_i\}_{i\in I}\) is a closed proper numerable covering of \(B\) for which each restriction \(f_i: p^{-1}(B_i)\to q^{-1}(B_i)\) is a proper homotopy equivalence over \(B_i\), then \(f\) is a proper homotopy equivalence over \(B\).NEWLINENEWLINEAll of the above is done in the more general framework of exterior homotopy theory where, along the way, useful results are obtained.