One-dimensional foliations on topological manifolds

Abstract

Let $X$ be an $(n+1)$-dimensional manifold, $\Delta$ be a one-dimensional foliation on $X$, and $p: X \to X / \Delta$ be a quotient map. We will say that a leaf $\omega$ of $\Delta$ is special whenever the space of leaves $X / \Delta$ is not Hausdorff at $\omega$. We present necessary and sufficient conditions for the map $p: X \to X / \Delta$ to be a locally trivial fibration under assumptions that all leaves of $\Delta$ are non-compact and the family of all special leaves of $\Delta$ is locally finite.