I am trying to show that if each $(X_\alpha, \tau_\alpha)$ is $T_2$ and $\Pi_{\alpha \in A} X_\alpha$ is complete, then each factor $X_\alpha$ is homeomorphic to a closed subspace of the product.
Let $\alpha_0\in A$ and consider $A_{\alpha_0}$, I was thinking of constructing the set $\Pi_{\alpha\in A} K_\alpha$ where $K_\alpha$ is a singleton when $\alpha \neq 0$ and $K_\alpha = X_\alpha$ when $\alpha = \alpha_0$. Certainly $A_{\alpha_0}$ is homeomorphic to this subset. But I am not sure if this set is closed ... is this the case?
On one hand I want to say it is closed since if you take a convergent net in the constructed set, it must converge inside the constructed set.
On the other hand I want to say it is not closed, since if you consider the complement, I can't fit a basic open set inside.
Hint: Take a point which is not in $\prod_\alpha K_\alpha$. Can you construct an open set around it not intersecting $\prod_\alpha K_\alpha$? (Maybe using the fact that your spaces are $T_2$...)