Consider the metric spaces (Xi,di),i=1,⋯,m. Check that d(x,y)=maxidi(xi,yi) defined on X=Πmi=1Xi is a metric. The metric topology coincides with the product topology on X. This may be concluded from the fact that for x=(x1,⋯,xm),Bd(x,r)=Πmi=1Bdi(xi,r).