I also believe it's important for the system to actually send out an email stating the ticket has been "closed", not just updated, if it has in fact been closed. This is because users may assume that the ticket is still being worked on if they haven't been specifically told that the ticket has been closed/completed
For now I just use the "post reply" and check the close button so that at least the user gets some kind of email notification, but a "ticket closed" notification would be nice!