A [standard problem of this type][1] is, can one draw uncountably many non-intersecting, non-degenerate figure-eights in the plane?  The problem is trivially "yes" for circles, rather than figure-eights, so I found this problem surprising when I first saw it. 


  [1]: http://books.google.com/books?id=ElSi5V5uS2MC&lpg=PA60&ots=_n8S5U40Mz&dq=figure%2520eights%2520in%2520the%2520plane%2520countable&pg=PA60#v=onepage&q=figure%2520eights%2520in%2520the%2520plane%2520countable&f=false