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Does a manifold without metric exist? If it exists, what is its name?

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- #1

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Does a manifold without metric exist? If it exists, what is its name?

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Topological manifolds (sets with a topology locally homeomorphic to R^{n}) do not necessarily admit a metric. There are then many non-metrizable manifolds, such as the Prüfer manifold. Urysohn's metrization theorem will let you know if your manifold is metrizable.

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About smooth manifolds: you can give a smooth manifold additional structure, perhaps by defining a Riemannian or Lorentzian metric tensor. As Halls hinted, as per the fundamental local versus global distinction in manifold theory, even after defining a Riemannian or Lorentzian metric tensor, there will be multiple distinct notions of "distance in the large" which may or not correspond roughly to the notion of "metric" fro m "metric space". In particular, Lorentzian metrics get their topology from the (locally euclidean) topological manifold structure, not from the bundled indefinite bilinear form.

(I'm being a bit more sloppy than usual due to PF sluggishness.)

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