# general topology – Open subset of a polish space

HERE is the theorem about being polish of open subsets of polish spaces. Let $$X$$ be a Polish space and the complete metric on $$X$$ is $$d<1$$ and the new one defined on any open set $$Usubset X$$ is as $$d^{*}(x,y)=d(x,y)+ |frac{1}{d(x,Xsetminus U)}-frac{1}{d(y,Xsetminus U)}|$$
clearly it is a metric on $$U$$ so it says that for each $$i$$, $$frac{1}{d(x_{i},Xsetminus U)}$$ is greater than $$1$$, so bounded away from zero, how is that $$d(x, Xsetminus U)>0$$ ($$x$$ is the limit of $$x_{n}$$ in $$X$$.