5/17    Would someone please explaine how a set can be both open and closed
        at the same time?  --PeterM
                \_ Being clopen is rare in Euclidean space with the usual
                   metric. The only clopen sets are the entire space
                   and the empty sets. However, when you endow the space
                   with some other topology, the you can get many clopen
                   sets . For example, the discrete topology implies that
                  *every* set is clopen. The key point is that open
                   and closed sets are not opposites, but simply classes
                   of sets with different definitions. Closed essentially
                   means "limits of sequences of points are in the set"
                   and open means "every point has a little fuzzy neigborhood"
                   around it. If you think of it that way, it's not too hard
                   to believe that under some topologies you can "trick"
                   a set into being open and closed at the same time. fab
        \_ open and closed seem like oposites most of the time, but their not.
           The only good example I can think of this is the whole line
      (-infty,infty) - has all of its limit points and yet still every point
      is in a neighborhood which is in the set.  'course that implies that
      the nought set is open and closed as well.  There are more funky
      examples when your universe isn't R.
           (-infty,infty) - has all of its limit points and yet still every
           point is in a neighborhood which is in the set.  'course that
           implies that the nought set is open and closed as well.  There are
           more funky examples when your universe isn't R.
           \_ You can usually find good examples of unintuitive results by
              considering a discrete metric space (i.e. a set of points
              with the metric d(a,b) = 0 if a = b and 1 otherwise).  In this
              space every set of points is closed because any convergant
              sequence of points in the set converges to a limit point in
              the set and open because an epsilon-ball centered on point X
              in the set only contains points in the set (in fact the
              epsilon ball only contains X). -emin
        \_ Both of the above posts are right. Keep in mind that "closed" is
           often defined as "complement of an open set", and what sets are
           open is part of the definition of the topological structure of
           the space (which has to obey certain axioms).