I (with some twist) described the set of 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 is a reparametrized integral curve for a direction field iff .
(Here and are certain funcoids determined by the vector field.)
You can understand this theorem after reading my research monograph.