A ticket mark "Resolved" will stay in your ticket view until you or the client "Close" the ticket.
Note: You can turn this off my navigating to Setup -> Tickets -> Status Types -> Choosing "Resolved" Status Type -> Uncheck the "Include in My/Group Tickets Filter" option. You can also choose to set the amount of time to "Automatically Close Ticket After" option to Auto Close after a said amount of time.
You can also apply a filter to your view to show just "Opened" tickets - thus purging your view of the "Resolved" (and all other status's) tickets.
I hope this helps.