Characterization of the Closure in a Metric Space by Open Balls

Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of all subsets of $X$ that are open in $(X,d)$, which is a topology on $X$ by Metric Open Sets Form a Topology. Let $A\subseteq X$ and let $x\in X$.

Then the following are equivalent.

1. The point $x$ belongs to the closure $\operatorname{cl}_X(A)$ of $A$ in the topological space $(X,\mathcal{T}_d)$.

2. For every real number $\varepsilon>0$, the open ball $B_d(x,\varepsilon)$ satisfies $B_d(x,\varepsilon)\cap A\neq\varnothing$.

3. For every real number $\varepsilon>0$ there exists $a\in A$ with $d(x,a)<\varepsilon$.