# Is any continuous curve in $\mathbb{R}^n$ a 1-D manifold?

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

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