Show that if $A_{\alpha}$ is closed in $X_{α}$, then $\prod A_\alpha$ is closed in $\prod X_\alpha$

Question

Show that if $A_{\alpha}$ is closed in $X_{α}$, then $\prod A_\alpha$ is closed in $\prod X_\alpha$.

Please could you help me? I have no idea how to do it. I appreciate any help, hint or solution. Thanks a lot.

Answer 1

The projections $p_\alpha: \prod_{\alpha \in A} X_\alpha \to X_\alpha$ are all continuous and so

$$\prod_{\alpha \in A} A_\alpha = \bigcap_{\alpha \in A} p_\alpha^{-1}[A_\alpha]$$

is an intersection of closed sets in $\prod_{\alpha \in A} X_\alpha$ and hence closed too.