Show that a subspace of a Hausdorff space is Hausdorff.
Solution
Let \(Y\subseteq X\), where \(X\) is Hausdorff, and let \(y_1\ne y_2\) be points of \(Y\). Choose disjoint open sets \(U_1,U_2\subseteq X\) with \(y_i\in U_i\). Then \[ U_1\cap Y,\qquad U_2\cap Y \] are disjoint open neighborhoods of \(y_1,y_2\) in the subspace \(Y\). Hence \(Y\) is Hausdorff.