On Fri, 12 Jul 2013, quasi wrote: > William Elliot wrote: > > >Let (S,d) be a compact Hausdorff space and f:S -> S a > >homeomorphism. How can it be show that there are some distinct > >x and y for which d(f(x),f(y)) = d(x,y)? > > Presumably you meant compact _metric_ space. > Still, I don't believe the claim.
Me neither. I left out an important premise. Let p1,.. pj be j consequitive points on a circle for which d(p1,p2),.. d(p_(j-1),pj), d(pj,p1) are all different and rotate the points, p1 -> p2,.. p_(j-1) -> pj, pj -> p1.