structural:reformatting, stylistic:tone
Euclidean space

A subset U {\displaystyle U} of the Euclidean n -space R is open if, for every point x in U {\displaystyle U}, there exists a positive real number ε (depending on x) such that any point in R whose Euclidean distance from x is smaller than ε belongs to U {\displaystyle U}. Equivalently, a subset U {\displaystyle U} of R is open if every point in U {\displaystyle U} is the center of an open ball contained in U. {\displaystyle U.}

An example of a subset of R that is not open is the closed interval, since neither 0 - ε nor 1 + ε belongs to for any ε > 0, no matter how small.