Section 30: Problem 12 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
We just need to show that the image of a basis is a basis of the image of the space. If
is a neighborhood of
then
is a neighborhood of
, and there is a basis open set containing
, its image is a basis neighborhood of
(as
is open).