Hello everyone,

I would really appreciate some help, as I'm stuck.

The following is a theorem from R. M. Dudley's Real Analysis and Probability.

First, the yellow. Theorem 13.1.5 that is mentioned states the following:

It says that there is a continuous function from onto B, not from any complete separable metric space. Does this follow somehow?

Then, the red. I don't see why would such neighborhoods exist.

Why would there even exist two closed disjoint neighborhoods of and ?

Could someone help?

Thank you.