I (with some twist) described the set of $C^1$ integral curves for a given vector field in purely topological terms (well, I describe it not in terms of topological spaces, but in terms of funcoids, more abstract objects than topological spaces).
Theorem $f$ is a reparametrized integral curve for a direction field $d$ iff $f\in\mathrm{C}(\iota_D|\mathbb{R}|_{>};Q_+)\cap\mathrm{C}(\iota_D|\mathbb{R}|_{<};Q_-)$.
(Here $Q_+$ and $Q_-$ are certain funcoids determined by the vector field.)