Equivalence relations on metric spaces

Question

Let $d:X\times X \rightarrow \mathbb{R} \cup \{\infty\}$ be a metric on the set X.

I should prove that $d(x,y)\neq \infty$ is an equivalence relation but I'm not sure what this expression means. $\neq$ is most certainly not an equivalence relation since it is not reflexive. Does it mean the set of all finite distances on $X$? What is the relation then?

Thank you.

Answer 1

You are asked to prove that the relation on $X$ given by $x \sim y \Leftrightarrow d(x, y) \not = \infty$ is an equivalence relation.

Concretely, that is:

• $d(x, x) \not = \infty$;
• $d(x, y) \not = \infty$ means $d(y, x) \not = \infty$;
• if $d(x, y) \not = \infty$ and $d(y, z) \not = \infty$, then $d(x, z) \not = \infty$.

Answer 2

Let the equivalence relation $\rho$ be given by $a\rho b\iff d(a,b)<\infty$

Now $d(a,a)=0\forall a\in X$ thus $\rho $ is reflexive.

Let $a\rho b$; then $d(a,b)<\infty\implies d(b,a)<\infty \implies b\rho a\implies \rho$ is symmetric.

Again let $a\rho b$, $b\rho c$ then $d(a,b)<\infty, d(b,c)<\infty$,By triangle inequality we have $d(a,c)\leq d(a,b)+d(b,c)<\infty\implies a\rho c\implies \rho $ is transitive