metric spaces notation

Question

Can \begin{equation} d(x,y) < 5 \end{equation} be written as \begin{equation} y \in U_5(x)\end{equation} ?

I am curious because I have seen $d(x,y) < \epsilon$ be written as $ y \in U_\epsilon(x)$.

Thank you.

Answer 1

Given the notation that $d(x,y)<\epsilon$ implies that $y \in U_{\epsilon}(x)$, your assertion is perfectly valid. But again, it is notation, so you could use whatever is most convenient to you. For instance, Rudin uses $N_{r}(x)$. The notation itself is arbitrary.

Answer 2

Probably, your definition of $U_r$ is $$U_r(x):=\{y\in X\mid d(x,y)<r\}$$ Therefore $y\in U_r(x)$ is equivalent to $d(x,y)<r$. This holds whatever variables (or constants) you insert for $x$, $r$ or $y$.