I want to find an example of metric $d$ for $[0,1]$ so that $(d,[0,1])$ is a separable metric space; $d$ is not equivalent to usual metric, but the Borel $\sigma$-field is same as the usual one.
Here the term "equivalent" means topological equivalence of metric. I found this problem in the book by Patrick Billingsley, "Convergence of Measures", Problem 1.7.
Here is what I found in the StackExchange. link. But no answer was appropriate fits, so I asked again.