How can I prove that $(X,τ)$ is a Hausdorff topological space?

Question

Let $(X_1,τ_1)$ is a Hausdorff topological space and $(X_2,τ_2)$ is a Hausdorff topological space and $X=X_1\times X_2$ and $τ$ The product topology

How can I prove that $(X,τ)$ is a Hausdorff topological space ?

Answer 1

Let $(x_1,x_2)$, $(x_1',x_2')$ be distinct points in $X_1\times X_2$. Then $x_1\ne x_1'$ or $x_2\ne x_2'$; without loss of generality we can assume the former. Since $X_1$ is Hausdorff, we may choose disjoint neighborhoods $U,V\subset X_1$ such that $x_1\in U$ and $x_1'\in U'$. If $x_2\ne x_2'$ then choose disjoint neighborhoods $V,V'$ of $x_2,x_2'$ in $X_2$; else set $V=V'=X_2$. Then $(x_1,x_2)\in U\times V$, $(x_1',x_2')\in U'\times V'$, and $$(U\times V)\cap(U'\times V') = (U\cap U')\times (V\cap V') = \varnothing, $$ so that $X_1\times X_2$ is Hausdorff.