Let (X,d) be a metric space.
a) Prove that for every x∈X, there either exists a δ>0 such that B(x,δ)={x}, or B(x,δ) is infinite for every δ>0.
b) Find an explicit example of (X,d), X infinite, where for every δ>0 and every x∈X, the ball B(x,δ) is finite.
c) Find an explicit example of (X,d) where for every δ>0 and every x∈X, the ball B(x,δ) is countably infinite.
d) Prove that if X is uncountable, then there exists an x∈X and a δ>0 such that B(x,δ) is uncountable.