discrete topology?

Question

Suppose that X is an uncountable set and d is any metric on X
which produces the discrete topology.
(such a metric d does not have to be a constant multiple of the discrete unit metric!)
Show that for some e>0 there is an uncountable subset A of X
such that d(x,y)>=e for all x,y in A.

Answer 1

First of all, there's a minor mistake on your assertion. Actually, we are asked to prove there's an uncountable subset A of X
such that d(x,y)>=e for all x,y in A such that x IS DIFFERENT from y. Right?

Around each point x of X, you can construct an open ball
B(x,r_x) where the radius r_x>0 is such that B(x, r_x) = {x) (this is because {x} must be open, since d induces the discrete topology on X and, therefore, X has no accumulation points). By construction, there's a bijection of X and the set {r_x) of all r_x, which implies {r_x} is uncountable.

Now consider the sets A_n=[1/n;+oo), where n is a
positive integer. We have U A_n = (0,+oo) (which contains all radii r_x), whence there exists N such that A_N contains uncountably many r_x. To see this, observe that the sets A_n form a countable collection whose union contains the whole uncountable set {r_x}. If all of the A_n`s contained only countably many radii r_x, then the same would be true of their union, contradicting the fact that this union contains all of {r_x}..

To finish the proof, it suffices to put e = 2/N and A = {x in X corresponding to the r_x in A_N}. A is uncountable, since it's equivalent to the uncountable set of the r_x in A_N. And by the choice of A_N, we must have d(x,y) >= 2/N = e for every x, y in A with x distinct from y (each element of A is enclosed in an open ball with radius r_x >= e/2 and this open ball equals {x}).

I hope I have helped. Email me if u have any question.

A nice problem!,