Selected Problems in Elementary Topology:

1) Construct example of a function  f: R -> R which is open but nowhere continuous. 
2) Prove that a subset  E  of  is the set of discontinuity of a function          f: R -> R  if and only if  E  is a countable union of closed subsets of  R
3) Let M be a closed connected orientable Riemannian manifold,  f: M -> M is a smooth map of nonzero degree which does not increase the metric. Show that  is an isometry.

More generally: assume only that  M  is a closed connected orientable topological manifold which is given structure of a geodesic metric space,      f: M -> M  is a distance-nonincreasing map of nonzero degree. Show that     f  is an isometry.