
 [No title] 
  Clearly, any homeomorphism will have to fix the point of attachment, but more is true: the points of the interval have the property that X{x} is not connected, but the points of the circle don't; yet all these points have the same kind of neighborhoods. 
  It's easy to find a homeomorphism of the unit ball carrying the origin to any other spot, and in fact to do so without moving points on the boundary of the ball. 
  I think it may be possible to find other examples based on, say, the Zariski topology, since typically an algebraic variety has only a finite number of selfisomorphisms (and thus in particular the orbit of a point under this family of maps cannot be the whole space). 
 www.math.niu.edu /~rusin/knownmath/95/homogen (1110 words) 
