Homeomorphism between a line segment and a subset of $\mathbb{R}^{2}$ that is not a curve I was just wondering if there is a homeomorphism between the line segment $[0, 1]$ and a subset of $\mathbb{R}^{2}$ that is not a curve. I would appreciate any eye-opening answers that would make it clear whether or not this is a possibility.
Thanks

The usual definition of a curve is (the image of) a continuous map from $[0,1]\to\mathbf{R}^2$. If $[0,1]$ and a subset $S\subset\mathbf{R}^2$ are homeomorphic, then in particular $S$ is the image of the homeomorphism. So the answer to your question is no; any subset of $\mathbf{R}^2$ that is homeomorphic to an interval is in fact a curve.
