Tosh,
Assuming XP.
Put a shortcut for Outlook in the start up folder.
C Documents and Settings\Tosh\Start Menu\Programs\Startup
Then right click the icon and in properties select run minimised.
Now when you start the PC the outlook icon will appear in the notification bar and it will be running all the time in the background.
You will get notifications of new mail.
The programme is not technically closed but it will do the job.
Sorry if this is obvious but I can take the flack!
MC