May I refer you to:
taken from page 92 of Infinite Dimensional Analysis by Charalambos D. Aliprantis.
Why $G \subset f^{-1}(W)$ ? I don't see this inclusion. Can you please help?
May I refer you to:
taken from page 92 of Infinite Dimensional Analysis by Charalambos D. Aliprantis.
Why $G \subset f^{-1}(W)$ ? I don't see this inclusion. Can you please help?
An alternative proof: Of course $Y = f[X]$ is compact and Hausdorff. It has a countable network (which is like a base of a space but without the requirement that its members are open sets), as a continuous image of a space with a countable network. A compact Hausdorff space with a countable network is second-countable (Arhangel'skij's theorem) and we are done by Urysohn. For a proof of the stuff on networks, I have a post here that explains it (at the end it goes into networks).