If and only if condition for a product space to be hausdorff.

Question

Is this true

$\forall i \in I \ X_i $ is hausdorff $\iff \prod_{i \in I}X_i $ is hausdorff.

I understand that $(\rightarrow )$ is true but don't know if $(\leftarrow)$ is true. If the other direction is true then is it true for both the box and product topology, if so how does one show this, if not true is there any counter example.(Where I is assumed to be a infinite index)

Answer 1

The projection map may not be good enough; the continuous image of a Hausdorff space is not necessarily Hausdorff.

Hint: You can find a copy of $X_n$ inside of the product. For each $i\in I$ let $p_i\in X_i$. Let $n$ be given. Consider $$Y_n:=\left\{(x_i)\in \prod X_i:x_i=p_i\text{ for }i\neq n\right\}\;.$$