Covering maps with separators (Q1803750)

Let \(\pi: E\to X\) be an \(n\)-fold covering map. A continuous map \(f: E\to S\) such that \(f(e_ 1)\neq f(e_ 2)\) whenever \(e_ 1\) and \(e_ 2\) are different points of the same fibre is called a separator for \(\pi\) with separation space \(S\). The use of various spaces as potential separators is investigated. For example each equivariantly polynomial \(\mathbb{Z}_ n\)- bundle admits \(S^ 1\) as a separation space; if \(X\) has the homotopy type of a connected CW-complex of dimension \(k\geq 1\) and \(M\) is a \((k- 1)\)-connected manifold of dimension \(> k\) then \(M\) is a separation space for every finite covering map of \(X\). The possibility of topological rings and fields being separators is also considered.