I have seen some questions and seen some idea. I am new to this topic and still find it confusing. I am kind of looking for a “algorithm “ to do them. I am using Sets & Metric spaces by Kaplansky Section 5.2 pgs 94-96,1977 printing
My understanding in order to prove them
If (M,d) is a metric space. To show its separable ,show it has a countable dense subset
Show some how that an open base exists for the given case.
Reading back in the book as to dense subset, ”...If A is dense in B then there exists a sequence $(x_i)$ -> x ”, Theorem 52,pg 89 I know it is talking about isometry.. It is the dense subset part that I am interested in
Looking at Theorem 58 I think I can do the easier ones, if I mimic Kaplansky’s proof style.