1617-257/TUT-R-8
On 11/3/16, we discussed some questions from the exam:
Problem 1. Let be an infinite subset of a compact metric space
. Show that
has a limit point.
Proof. If has no limit points, then
is a closed subset of a compact space and is therefore compact in itself. Since each point of
is isolated, we may find for each point
a neighborhood
such that
. The collection
is an open cover of E which clearly has no finite subcover.
Problem 2. Let be a function which is "jelly-rigid": for all
:
. Prove that
maps onto
.
Proof. Since is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on
. A simple estimate shows if
, then
. Suppose now that there is some point
which is not in the image of
. Let
be a closest element to
in the image of
. Consider now the point
(jelly-rigidity says that the function is almost like the identity, so moving closer to the point
from
in the codomain side can be obtained by moving in that same direction on the domain side first) where
is chosen to be small enough so that
and
. Then
is closer to
than is
.