In a

metric space, a set U is an

open subset of X if for every point x in the set, there exists some positive ε such that B(x,ε)={x in X | |x|;<ε} is contained in U.

For example, in the real numbers the set (0,1) = {numbers between 0 and 1, not including either} is open, whereas [0,1] = {numbers between 0 and 1, including both} is not (ie. because no such positive ε exists for 1 or 0).

This definition gives rise the properties that:

...which become the defining properties of open sets in

topological spaces.