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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.

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

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    $\begingroup$ 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 $\endgroup$
    – Andrew Clifford
    Commented Jul 12, 2012 at 10:57
  • $\begingroup$ 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. $\endgroup$
    – Aru Ray
    Commented Jul 12, 2012 at 12:35

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