Thread: two questions about a proof of a theorem on Borel spaces

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.

Originally Posted by harriette

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.

I am not sure I can help, take everything I can say with a grain of salt. But, isn't the red true because this is normal?