I am not using 'ball' to describe a neighbourhood, you are using 'neighbourhood' to describe a ball! From your own link: In a metric space M=(X,d), a set V is

I am not using 'ball' to describe a neighbourhood, you are using 'neighbourhood' to describe a ball! From your own link: In a metric space M=(X,d), a set V is a neighbourhood of a point p if there exists an open ball with centre p and radius r>0, such that B_r (p)=B(p;r)={x∈X | d(x,p)<r} is contained in V. (My emphasis.)