Prove that the product topological space \(X\times Y\) is Hausdorff if and only if \(X\) and \(Y\) are Hausdorff.
Solution
The implication from “\(X\times Y\) is Hausdorff” to “\(X\) and \(Y\) are Hausdorff” requires \(X\) and \(Y\) to be nonempty: if \(Y=\varnothing\), then \[ X\times Y=\varnothing \] is Hausdorff even when \(X\) is not.
1. If \(X\) and \(Y\) are Hausdorff, then \(X\times Y\) is Hausdorff.
Take distinct points \[ (x_1,y_1),(x_2,y_2)\in X\times Y. \] Since they are distinct, either \[ x_1\ne x_2 \] or \[ y_1\ne y_2. \]
Suppose first that \(x_1\ne x_2\). Since \(X\) is Hausdorff, there are disjoint open sets \(U_1,U_2\subseteq X\) with \[ x_1\in U_1, \qquad x_2\in U_2. \] Then \[ U_1\times Y \qquad\text{and}\qquad U_2\times Y \] are disjoint open neighborhoods of \((x_1,y_1)\) and \((x_2,y_2)\) in the product topology.
If instead \(y_1\ne y_2\), choose disjoint open neighborhoods \(V_1,V_2\subseteq Y\) of \(y_1,y_2\) and use \[ X\times V_1 \qquad\text{and}\qquad X\times V_2. \] Thus any two distinct points of \(X\times Y\) have disjoint open neighborhoods, so the product is Hausdorff.
2. Every subspace of a Hausdorff space is Hausdorff.
Let \(A\) be a subspace of a Hausdorff space \(Z\), and take distinct \(a,b\in A\). Choose disjoint open sets \(U,V\subseteq Z\) with \[ a\in U, \qquad b\in V. \] Then \[ U\cap A \qquad\text{and}\qquad V\cap A \] are disjoint open neighborhoods of \(a,b\) in the subspace \(A\). Hence \(A\) is Hausdorff.
3. Assume \(X\ne\varnothing\), \(Y\ne\varnothing\), and \(X\times Y\) is Hausdorff.Then \(X\) is Hausdorff.
Choose \(y_0\in Y\). The map \[ i_X:X\longrightarrow X\times\{y_0\}, \qquad i_X(x)=(x,y_0), \] is a homeomorphism: it is continuous, and its inverse is the restriction of the first coordinate projection \[ p_X:X\times Y\to X. \] By 2, the slice \(X\times\{y_0\}\) is Hausdorff as a subspace of the Hausdorff product. Hausdorffness is preserved by homeomorphism, so \(X\) is Hausdorff.
4. Under the same assumptions, \(Y\) is Hausdorff.
Choose \(x_0\in X\). The map \[ i_Y:Y\longrightarrow\{x_0\}\times Y, \qquad i_Y(y)=(x_0,y), \] is a homeomorphism whose inverse is the restriction of the second coordinate projection. The slice \(\{x_0\}\times Y\) is Hausdorff by 2, so \(Y\) is Hausdorff.
5. Therefore, for nonempty spaces \(X\) and \(Y\),\[ \boxed{X\times Y\text{ is Hausdorff}\iff X\text{ and }Y\text{ are Hausdorff}.} \]
The forward implication follows from <1>3–<1>4, and the reverse implication is 1.