Why are homotopies defined by real intervals?

Question

In homotopy theory, a homotopy between two functions $f\ g : A \to B$ is a function $X : [0, 1] \to A \to B$ such that $X(0) = f$ and $X(1) = g$. How do we know that the real interval $[0, 1]$ captures the right notion of continuity? I find it unintuitive that such an elementary concept necessarily depends on the real numbers, which seem much less fundamental than homotopies. From the perspective of homotopy type theory, this seems like a strange dependency.

Answer 1

It is ordered, connected, and has both endpoints, rather desirable properties. See https://mathoverflow.net/a/123797/13113 for a full characterization.

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An alternative characterization comes from https://math.stackexchange.com/a/1824082/14972 that considers generalized paths from compact Hausdorff spaces with two marked points, and shows that the interval is the minimal universal path.