structural:expansion, linguistic:paraphrasing
Metric space

A subset U of a metric space (M, d) is called open if, for any point x in U, there exists a real number ε > 0 such that any point y ∈ M {\displaystyle y\in M} satisfying d (x, y) < ε belongs to U. Equivalently, U is open if every point in U has a neighborhood contained in U.

This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.