I (with some twist) described the set of $latex C^1$ 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).

From this PDF file:

**Theorem** $latex f$ is a reparametrized integral curve for a direction field $latex d$ iff $latex f\in\mathrm{C}(\iota_D|\mathbb{R}|_{>};Q_+)\cap\mathrm{C}(\iota_D|\mathbb{R}|_{<};Q_-)$.

(Here $latex Q_+$ and $latex Q_-$ are certain funcoids determined by the vector field.)

You can understand this theorem after reading my research monograph.