For a vector $\vec w$, let $T_{\vec w}$ be the translation through $\vec w$.

I was told that the following observation about subsets of the plane was due to H. Hopf:

Let $X$ be a compact, path-connected subset of the plane. Then, if for some vector $\vec v$, $X\cap T_{\vec v}[X]\neq \emptyset$, then for each positive natural number $n$, $X\cap T_{\frac{1}{n} \vec v}\neq \emptyset$.

I have a proof (which may or may not be Hopf's), but what I want is the reference to cite.

My understanding is that Hopf used is to prove that if a 1-sphere properly embedded on the torus represents the homology element $al+bm$ (where $l$ and $m$ are the longitude and meridian) then $a$ and $b$ are relatively prime.

Thanks in advance

Andrew Clifford