In $\mathbb{R}^2$ closed sets are precisely the zero sets for continuous functions from $\mathbb{R}^2$ to $\mathbb{R}$.
Which closed sets can not be curves? Of course closed subsets which contain a closed disc or rectangle are not curves.
I came to following question:
Let $C$ be the collection of those points $(x,y)$ where $y=\pm 2^nx$ for $n\in\mathbb{Z}$, along with $y$-axis. Can it be a curve?
I was thinking that the point $(0,0)$ will make some problem to declare it to be a curve, but I could not justify it precisely.