Question

CUISI
b. The student has to complete the assignment in the allocated pages only. Any other page in case utilized shall
Q2 Prove that every continuous image of a separable space is separable. [10 Marks]
not be considered.​

Answer 1

Answer:

Let (X,T) be a separable space. Then there exists some countable subset A⊆X such that A¯¯¯¯=X. Let f:X→Y be a continuous mapping. Notice that f:X→f(X)⊆Y is clearly surjective.

Since A⊆X, and functions preserve set inclusion, we have that

f(A)⊆f(X)=f(A¯¯¯¯).

Also, it is clear that f(A) is itself also countable.

What (I think) I need to show, however, is that f(A)¯¯¯¯¯¯¯¯¯¯=f(A¯¯¯¯) (which I am not sure if it's true). This will thus show that f(A) is a countable, dense subset of f(X).