Section 30: 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
If
is a continuous map then the dense maps to the dense (each open set contains the image of a dense point in the preimage of the set) and for any covering of the image a subcovering may be obtained as the image of a subcovering of the preimage of the collection. Therefore, if the space is separable, so is the image, and if the space is Lindelöf, so is the image.