Equivalence of definitions of open sets in metric space Consider the following two equivalent definitions of an open set in a metric space $(X,d)$:

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*Definition 1: A set $U$ in a metric space is open if and only if $U$ is an arbitrary union of open balls of elements in the metric space, or an open ball itself.


*Definition 2: A set $U$ in a metric space is open if and only if, for every element $x\in X$, there exists an $\epsilon$ such that an open ball $B_{\epsilon}(x)$ is contained within $U$.
Although Definition 2 clearly implies Definition 1, how does Definition 1 imply Definition 2?

Definition 1 implies 2 because of the observation that if $B(x,r)$ is an open ball and $y\in B(x,r)$ then for $r'=r-d(x,y)$ (which is positive) we have $B(y,r')\subseteq B(x,r)$. The last inclusion follows from the triangle inequality.
