Exercises
[0R2] Suppose that \(d\) satisfies all distance requirements except ”separation property”. Consider the relation \(∼\) on \(X\) defined as \(x∼ y\iff d(x,y)=0\); show that is an equivalence relation. Let’s define \(Y=X/∼\); show that the function \(d\) “passes to the quotient”, that is, there exists \(\tilde d:Y× Y→ [0,∞)\) such that, for every choice of classes \(s,t∈ Y\) and every choice of \(x∈ s,y∈ t\) you have \(\tilde d(s,t)=d(x,y)\). Finally, show that \(\tilde d\) is a distance on \(Y\).
This procedure is the metric space equivalent of Kolmogoroff quotient.