# Hopf reference sought

For a vector $w$, let $T_{w}$ be the translation by $w$.

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 $v$, $X\cap T_{v}[X]\neq \emptyset$, then for each positive natural number $n$, $X\cap T_{\frac{1}{n} \cdot v} \neq \emptyset$.

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.

## 1 Answer

Check out the Chord Theorem (Theorem 2B12) in Rolfsen's Knots and Links. Here is a link to the Google book.

• Thanks. That's a very good reference. Rolfsen doesn't mention Hopf there, so I wonder if it really is due to Hopf. Andrew Clifford Jul 12 '12 at 10:57
• I expect this to be one of those results which are so basic at this point that one may either leave them uncited or cite some basic textbook. Jul 12 '12 at 12:35