I have a small trouble while trying to grasp which fact is described by the following statement:
"If a set X has two different elements, then the indiscrete topology on X is NOT of the form for some metric d on X. Why? In particular, not every topology comes from a metric."
1. Now I tend to interpret the above statement that whenever X has (at least) two different elements and d is a metric on X, then . Then it makes sense to see a set with at least two elements as an example where the topology induced by a metric does not produce the empty set and the whole of X as the only open sets.
2. But the word some disturbs me. Because it allows for another interpretation, namely: whenever X has at least two elements there exists a metric d such that .
So before I head off to answer "why?" I would like to know: What do they mean exactly?