Is it possible to update an application to a new version without closing it?
Or is there a good way to do that without user noticing it was closed?
Typically applications notice on startup that an update is available, then ask the user whether it's okay to update. They then start the update process and exit. The update process replaces the files, then launches the new version.
In some cases you may be able to get away with updating some pieces of an application without a restart - but the added complexity is significant, and frankly it's better not to try in 99% of cases, IMO.
Of course, you haven't said what kind of app you're writing - if you could give more information, that would help.