Section 23: Problem 10 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 an indexed family of connected spaces; let
be the product space
Let
be a fixed point of X.
(a) Given any finite subset
of
, let
denote the subspace of
consisting of all points
such that
for
. Show that
is connected.
(b) Show that the union
of the spaces
is connected.
(c) Show that
equals the closure of
; conclude that
is connected.
(a) It is homeomorphic to a finite product of connected spaces.
(b) They all have a common point (
).
(c) This is the main part of the proof. Take any point
. Any its neighborhood in the product topology has only finite number of coordinates restricted by some proper open subsets of the corresponding coordinate spaces. A point that equals
at these finite number of coordinates, and equals
at all others coordinates, belongs to the neighborhood and to
. Therefore, the closure of
is the whole space
, and by Theorem 23.4,
is connected.