Selected Problems in Elementary Topology:
1) Construct example of a function f: R -> R which is open
but nowhere continuous.
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.
2) Prove that a subset E of R
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 f is an isometry.