Show that a product of two Hausdorff spaces is Hausdorff.
I rephrase this exercise to my taste.
If $(X,\mathcal{T}_{X})$ & $(Y,\mathcal{T}_{Y})$ are Hausdorff space, then $(X\times Y,\mathcal{T}_{p})$ is Hausdorff.
My attempt: let $\mathcal{B}=\{ U\times V| U\in \mathcal{T}_X, V\in \mathcal{T}_Y \}$ be a basis for $\mathcal{T}_{p}$. Let $x\times y, p\times q \in X\times Y$ such that $x\times y\neq p\times q$. Which implies $x\neq p$ or $y\neq q$. If $x \neq p$. Since $X$ is Hausdorff, $\exists M, N \in \mathcal{T}_X$ such that $x\in M$, $p\in N$ and $M \cap N= \phi$. So $M\times Y,N\times Y \in \mathcal{B}\subseteq \mathcal{T}_{p}$ with $x\times y \in M \times Y$ and $p\times q \in N \times Y$. We have $(M \times Y) \cap (N \times Y)=(M \cap N) \times Y=\phi \times Y=\phi$. If $y\neq q$. Proof is similar. Is this proof correct?
Is there rigorous way to prove $\phi \times \phi =\phi$?
