I think it would really help if you get notified that a question of yours was closed.

Especially for the multi-site power user, usually (at least for me) a question is asked and then left alone for while. I rely on the network-wide inbox to tell me when someone has commented or answered, but if I don't receive any messages then I tend to assume that the question wasn't answered or even noticed. It's only when I check the question a day or two later to see what I can add do I find out it was closed.

This would be especially helpful for duplicates, where the answers you need are already there. A message to the effect of, "Closed as duplicate of question" is just like notifying me that a new answer has been added!

Unfortunately, question closures can remain unnoticed by their askers until months have passed. By then, the questions may either be too old to migrate to the correct site, or may run the risk of automatic deletion, depending on the specific closing reason. We should notify users of these question closures so that they can improve them and learn from their mistakes.

Could closed questions get added to the inbox?