How many metrics are defined on a singleton set?

Question

I am a university student and am doing a first course in metric spaces. I came across this question in a past paper and was not sure if my reasoning is true.

Let S be a set, and let $M_S$ denote the collection of metrics on S. Suppose S is a singleton set, say $\{0\}$. How many metrics are on S?

My trail of reasoning is that since $\{0\}$ is a metric space then any metric defined on $\mathbb{R}$ can be applied to it. Since there are an infinite amount of metrics on $\mathbb{R}$ this implies $M_S = \infty$.

Thank you in advance.

Answer 1

If I understood correctly, you have a set $S=\{*\}$ with one element and your question is how many metrics can you define. A metric here will be a function $d:S\times S \rightarrow \mathbb{R}_{\geq 0}$ satisfying the axioms. Since there is only one pair of points you can evaluate, you must get $d(*,*)=0,$ so you should be able to conclude how many metrics are there.

Answer 2

Yes, there are infinely many matrics on $\mathbb R$ but it could happen that they are all equal when restricted to $\{0\}$. And, in fact that is exactly what happens! The only metric that you can define on $\{0\}$ is the function $d$ such that $d(0,0)=0$.