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On a few occasions, I've noticed that I haven't gotten any kind of notification when my own questions was closed. For example, when this question was closed as "off-topic", I didn't see any kind of notification in the Stack Exchange navigation bar, and I only noticed the change after I did a Google search for the question, and realized that it had been closed earlier.
I think users should be notified when their questions are closed so that they will at least be given a chance to improve their questions. Since no notification is given for question closures, some question closures continue to unnoticed 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.
I think notifications for closed questions should be implemented for all of the above reasons. The original poster of a question ought to be given a chance to improve and reopen their question, before it's too late for them to make any changes.