'Closed' is the final status of a ticket. Industry best practice dictates that you let the requester close the ticket after the agent changes the status to 'Resolved', to indicate the requester’s agreement with the solution. If they feel the problem hasn’t been solved, they can just reply to your ticket and move it out of the 'Resolved' pile.
The email notification sent out to customers when agents close the ticket is the best time to send out a satisfaction survey and find out what your customers feel about your support.
In most cases, customers either forget to close the ticket or have no idea that they’re supposed to do it. This means that the resolved but unclosed tickets could pile up in your helpdesk. To avoid this, you should consider setting up a business rule that closes all resolved tickets after a certain amount of time has passed (preferably after 48 hours).