A resolution for the product of a compactum with a polyhedron. (Q1403831)

Any compact Hausdorff space \(X\) can be represented as the inverse limit of compact polyhedra. For more general spaces one needs { resolutions} of a space, generalizing the concept of an inverse limit. The author deals with the following problem: Suppose \(\pmb p: X \to \pmb X\) is the limit of an inverse system of compact CW-spaces. Let \(K\) be a (not necessarily finite) simplicial complex, \(\mid K \mid =P\) its realization, is it true that \(\pmb p \times 1_P: X \times P \to \pmb X \times P\) is a resolution? The answer is \textit{no}: Take for \(P\) the discrete space of cardinality \(\aleph_0\). To repair this defect, the author associates with \(\pmb p\) and a simplicial complex \(K\) a so-called {basic construction}, yielding a resolution \(\pmb q: X \times P \to \pmb Y\) with \(\pmb Y\) consisting of paracompact spaces, having the homotopy type of polyhedra. It turns out, that \(\pmb X \times P\) becomes a subsystem of \(\pmb Y\), so that \(\pmb q\) extends \(\pmb p \times 1_P: X \times P \to \pmb X \times P\). The main theorem of the present paper asserts the existence of such a {basic construction}. Moreover the author treats the question how this basic construction behaves under subdivisions of \(K\) (observe that the basic construction depends not only on the space \(P\) but also on the simplicial complex \(K\)). The proof of the main result is rather involved and requires among other things, extensive use of the theory of cofibrations.