Product of Hausdorff Factor Spaces is Hausdorff

Theorem
Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces.

Let $T = T_\alpha \times T_\beta$ be the product space of $T_\alpha$ and $T_\beta$

Let $T_\alpha$ and $T_\beta$ both be $T_2$ (Hausdorff) spaces.

Then $T$ is also a $T_2$ (Hausdorff) space.

Proof
Let $X$ and $Y$ be Hausdorff topological spaces. Suppose (a,b), (c,d) are two disjoint points of the product space $X$ × $Y$. If a = c, then b ≠ d therefore ∃ two disjoint open sets U, V ⊆ $Y$ s.t. b ∈ U, d ∈ V. Then the $X$ × U and $X$ × V are two open disjoint sets in the product space $X$ × $Y$ containing the original points. If a ≠ c, then ∃ two disjoint open sets U, V ⊆ $X$ s.t. a ∈ U, c ∈ V. Then U × $Y$ and V × $Y$ are disjoint open sets in $X$ × $Y$ containing the original points.