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__Selected Problems in Elementary Topology:__

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**1) Construct example of a function ***f: R -> R* which is open
but nowhere continuous.

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.

**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.

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