Are there sets where it cannot possibly have a metric on it?

Question

To avoid any ambiguity, a metric space, by definition, is a set $X$ with a distance function $d$ such that $d$ satisfies positivity, symmetry property and triangle inequality.

I was wondering does there exist a set where there cannot possibly be equipped with a distance function? In other words this set cannot possibly be made into a metric space?

I hope I explained my question sufficiently clear and apologies in advance if this question was not clear. Many thanks in advance!

Answer 1

Any set can have a metric, because the discrete metric can be applied to all sets.

See here and here for further details about it.

Answer 2

No, there is no such set. You can always define the discrete metric on any set.

Answer 3

Every set can become a metric space.

Let $X$ be a set. Define $d: X\times X\rightarrow \mathbb{R}$ by $d(x,y)=0$ if $x=y$ and $1$ otherwise.

This is called the discrete metric on $X$.