Section 24: Problem 11 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
is connected in
and its interior is connected but its boundary is not. Two balls in
touching at a common boundary point is a connected set (as the union of two connected sets with a common point) and its boundary is connected (for the same reason), but its interior is not connected (as the union of two disjoint open sets). At the same time
has connected interior and boundary but is not connected.