Show that the product of two Hausdorff spaces is Hausdorff.
Solution
Let \(X\) and \(Y\) be Hausdorff, and let \(p_1=(x_1,y_1)\) and \(p_2=(x_2,y_2)\) be distinct points of \(X\times Y\), so that \(x_1\ne x_2\) or \(y_1\ne y_2\).
1. If \(x_1\ne x_2\), then \(p_1\) and \(p_2\) have disjoint open neighborhoods.
Choose disjoint open \(U_1\ni x_1\) and \(U_2\ni x_2\) in \(X\). Then \(U_1\times Y\) and \(U_2\times Y\) are open in \(X\times Y\), contain \(p_1\) and \(p_2\), and \((U_1\times Y)\cap(U_2\times Y)=(U_1\cap U_2)\times Y=\varnothing\).
2. If \(y_1\ne y_2\), then \(p_1\) and \(p_2\) have disjoint open neighborhoods.
Choose disjoint open \(V_1\ni y_1\) and \(V_2\ni y_2\) in \(Y\); then \(X\times V_1\) and \(X\times V_2\) are disjoint open neighborhoods of \(p_1\) and \(p_2\), as in step 1.
3. Q.E.D.
Steps 1 and 2 cover both cases.