Converse statements regarding separable metric space

Question

These statements:

a. Every infinite subset has a limit point

b. Separable

c. Has a countable base

d. Compact

are mentioned in the exercises of baby Rudin Chapter 2

I have proven with hints (in exercise 2.23) $b\Rightarrow c$, (in exercise 2.25) $d\Rightarrow c\Rightarrow b$, (in exercise 2.24) $a\Rightarrow b$, (in exercise 2.26) $a\Rightarrow d$ and finally (in theorem 2.37) $d\Rightarrow a$.

Now in order to make all these statements equivalent to each other, I have to prove either $b\Rightarrow a$ or $c\Rightarrow d$, but haven't succeeded in either. I wonder since Rudin left out these two statements intentionally, are they actually true? If they aren't what are some counterexamples?

Can somebody verify if these statements are indeed equivalent to each other in a metric space?

Much appreciated

Answer 1

Neither of those implications hold. Here are counter-examples:

a. $\implies$ b.: $\mathbb Q$ is separable, but $\mathbb Z$ has no limit poins.

c. $\implies$ d.: the topology of $\mathbb Q$ has a countable base, but $\mathbb Q$ is not compact.

Answer 2

b. and c. are equivalent in all metric spaces. $\Bbb R$ is separable in the usual metric but does not satisfy a and d. Statements a. and d. do imply separability for metric spaces. So they are not equivalent for all metric spaces. These are all the implications we can state (also d implies a for all spaces, as c implies b too also always holds).