Sets of integral curves described in topological terms

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).

From this PDF file:

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.)

You can understand this theorem after reading my research monograph.

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