Is it possible to find a smooth manifold on which it is impossible to define a metric function?
2026-03-28 22:25:24.1774736724
Non-metrizable smooth manifold?
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It depends on the definition of a smooth manifold $M$. Usually one requires that $M$
1) is Hausdorff,
2) is second countable,
3) has a smooth atlas.
The "minimal" requirement for a smooth manifold would be 3), but obviously 1) is a necessary condition for the existence of a metric. See https://en.wikipedia.org/wiki/Non-Hausdorff_manifold for examples of non-Hausdorff manifolds.
That 2) is necessary for the for the existence of a metric is less obvious. As a counterexample take the long line https://en.wikipedia.org/wiki/Long_line_(topology).
If 1) - 3) are satisfied, then deb's and Aleksandar Milivojevic's comments show that $M$ is metrizable.