Yes.
True or false. In an infinte metric space, an open ball of radius 2 is an infinte set.
Well, my answer is false,
Let M= R.
Let the metric be given by :
d(x,y) = 0 if x=y
d(x,y) = 2 if x<>y
Then in this metric space, the open ball B[a;r] about a in M has just a single point.
Is my reasoning right??