Hausdorff reductions of leaves spaces (Q2151140)

The author considers the following situation: a continuous map \(F:X\to M\) from a locally path connected topological space \(X\) to a topological space \(M\). The map \(F\) induces various quotients of \(X\); the author considers the space of leaves, where \(x\) and \(y\) are equivalent (\(x\sim y\)) if there is a path that connects \(x\) and \(y\) on which \(F\) is constant. Thus a leaf of \(F\) is a path component of some fiber of \(F\). The resulting quotient is denoted \(\tilde X\).\par There is a further equivalence relation \(\wedge\) on \(X\), the smallest one that contains \(\sim\) and satisfies: \textit{if} \(x\wedge y\) and there are paths \(\gamma_1,\gamma_2:[0,1]\to X\) such that \(\gamma_1(0)=x\), \(\gamma_2(0)=y\), and \(F(\gamma_1(t))=F(\gamma_2(t))\) for all \(t\) \textit{then} also \(\gamma_1(1)\wedge\gamma_2(1)\). The resulting quotient space is denoted \(\widehat X\).\par The author presents situations where \(\widehat X\) is Hausdorff and where the natural map between \(\tilde X\) and \(\widehat X\) is a local homeomorphism. The final part contains applications of these ideas to maps between manifolds and Riemann surfaces.