Reconciling metric and topological neighborhoods

Question

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?

Answer 1

Given a point in $x\in X$, an open neighborhood is more appropriately called an $\epsilon$-ball ...

That's not how I read the definition, e.g. Wikipedia's (in the "In a metric space" section). A neighborhood of a point must contain an $\epsilon$-ball around the point - it doesn't have to be an $\epsilon$-ball centered on the point. And a set is said to be open if it is a neighborhood for all the points out contains.

I don't see any contradiction that needs reconciling. Can you be more specific - is there a contradiction you still see, if you use this definition of neighborhood?