Section 53: Problem 3 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Let
be a covering map; let
be connected. Show that if
has
elements for some
, then then
has
elements for every
. In such a case,
is called a k-fold covering of
.
Sets
are open in
, as for
there is a neighborhood
such that
is the union of
disjoint open subsets
,
, of
such that
is a homeomorphism of
with
, hence, same sets work for any point in
, showing that the preimage of any point in
has
elements as well. Since
is connected, there must be only one such set
.