Let X be a set and d, d′ be two metrics on X. Suppose there exists an α>0 and β>0 such that αd(x,y)≤d′(x,y)≤βd(x,y) for all x,y∈X. Show that U is open in (X,d) if and only if U is open in (X,d′). That is, the topologies of (X,d) and (X,d′) are the same.