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I wonder if there is any theorem stating that any continuous curve in $\mathbb{R}^n$ is a 1-D manifold.

If not, can anyone provide an example?

At first I thought maybe a Peano curve affords a counterexample, but it seems not...

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What is your definition of a manifold? And think about the curve having self-intersection. – ronno Oct 29 '12 at 14:03
up vote 1 down vote accepted

There are different answers, depending what exactly you mean, but all are no in last consequence.

If with manifold, you mean SUBmanifold, the answer is no. Just think of a line with corners (ok, that one could still be a topological submanifold) or a line that intersects itself.

If you ask if every image of a curve is a manifold when equipped with the subspace topology, a plane filling curve provides a counterexample. This is because the neighborhood of any points has infinitely many connected components and can therefore not be homeomorphic to any Rn. An even simpler counterexample is yet again a self-intersecting curve.

However, all these curves can be immersed submanifolds, but that is rather trivial and mostly useless.

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Oh yes. I misunderstood the definition. Thx! – hxhxhx88 Oct 30 '12 at 13:00

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