I am learning about metric spaces.
I failed to find an aunambiguous answer to my question on Google. So this is the right place to ask:
Assume (X,d) is a metric space, and $A \subset X$. If I understand correctly, as long as $(X,d)$ is a metric space $\implies X \neq \varnothing$? If that is the case, then the following may or may not be true:
«$A$ is a strict subset of $X$ $\implies$ $A$ has points $x \in X : \forall \varepsilon > 0, \,\exists B(x,\varepsilon) : B(x,\varepsilon)\cap A \neq \varnothing \wedge B(x,\varepsilon)\cap A^c \neq \varnothing$»
If you know, will you kindly tell me if this is true, and why that is.
Thank you for your time.