Section 23: Problem 5 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
A space is totally disconnected if its only connected subspaces are one-point sets. Show that if
has the discrete topology, then
is totally disconnected. Does the converse hold?
If
and
, then
and
form a separation of
unless
. The converse does not hold: see Example 4.