Hausdorff Reductions of Leaves Spaces

Let F be a continuous mapping between topological spaces X and M and let x₁ ∼ x₂ if and only if these points belong to the same path connected component of F⁻¹(F(x₁)) be an equivalence relation on X. Under the assumption that the quotient mapping is open we construct a weaker equivalence ∧ on X and show that if we impose on F the lifting property, then the space X/∧ is Hausdorff. Replacing the lifting property with a stronger regularity property we show X/∧ is a manifold of the same class as M. Moreover, the construction of ∧ shows that X/∧ is the maximal manifold that can be obtained by weakening ∼. Finally we apply our results to holomorphic functions F on a complex manifold X.