Let $X$ be a metric space. Given a point in $x \in X$, an open neighborhood is more appropriately called an $\epsilon$-ball $N_\epsilon = \{p \in X : d(p, x) < \epsilon\}$, while a topological neighborhood is any open set containing $x$. I am wondering if there is a rigorous way to reconcile these definitions, based on the facts that open sets are unions of open balls. One example is in the definition of limit points: $p$ is a limit point of a subset $E \subseteq X$ iff for all open subsets $G \subseteq X$ such that $p \in G$, $E \cap (G - \{x\}) \ne \emptyset$. But can the definitions be used interchangeably in arbitrary formulas? If so, how to formulate this?
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