The open ball in a metric space ( X , d ) {\displaystyle (X,d)} with radius ϵ {\displaystyle \epsilon } centered at a, is denoted B ( a , ϵ ) {\displaystyle B(a,\epsilon )} . Formally B ( a , ϵ ) = { x ∈ X : d ( a , x ) < ϵ } {\displaystyle B(a,\epsilon )=\{x\in X:d(a,x)<\epsilon \}}
Let ( X , d ) {\displaystyle (X,d)} be a metric space. We say a set A ⊂ X {\displaystyle A\subset X} is open if for every x ∈ A ∃ ϵ > 0 {\displaystyle x\in A{\text{ }}\exists \epsilon >0} such that B ( x , ϵ ) ⊂ A {\displaystyle B(x,\epsilon )\subset A} .
We say a set B ⊂ X {\displaystyle B\subset X} is closed if X ∖ B {\displaystyle X\backslash B} is open.
