When An Infinite Product Topology Is Hausdorff?

The issue is, that $(x_1,x_2,...)$ and $(y_1,y_2,...)$ for $x_1\neq y_1$ is not general enough to represent every possible point.

Let $\tau_2 = \{X_2,\emptyset\}$ the trivial topology. Consider the points $x=(x_1,a,x_3,x_4,...)$, $y=(x_1,b,x_3,x_4,...)$. Pick any two open sets $A, B\in\tau$ such that $x\in A$, $y \in b$. Projected on their second component they have to be $X_2$. But then $x$ and $y$ lie at their intersection.