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May I refer you to: alt text

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?

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    @Arturo Magidin: Thank you too for adding it to the question. It's much better now.2011-01-05

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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).