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.
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.
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.
Probably, your definition of $U_r$ is $U_r(x):=\{y\in X\mid d(x,y)