Hartshorne proposition II(2.6)

Let $U=X\setminus Y$ be an open subset of $X$ (of course $Y$ is closed). The induced map can be defined as $$ \alpha(U) = t(X)\setminus t(Y), $$ which is well defined since $\alpha(U)$ is an open subset of $t(X)$ by definition of its topology. This map is obviously a bijection between the open subsets of $X$ and the open subsets of $t(X)$.