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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.

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Where'd the quantifier come from? It should just be $y \in U_{\epsilon}(x)$. – Qiaochu Yuan Dec 18 '12 at 6:06
Yes, ill edit that. – MathMathMath Dec 18 '12 at 6:10
Yes; $U_{r}(x)$ would mean "ball of radius $r$ around $x$." So $y\in U_{r}(x)$ exactly when $d(x,y) < r$. So this is just two different ways of expressing the same thing. – Deven Ware Dec 18 '12 at 6:18
up vote 2 down vote accepted

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.

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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 $y$.

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