Managing personal Outlook.com account settings, security, and privacy
>> Is browser open? I don't understand the question.
You are using Outlook on the web correct? (This is the outlook on the web forum.)
If so, go to Outlook.com's settings: Settings (gear icon) > View all Outlook settings > General > Notifications to turn the browser notifications off.
If you are using Outlook desktop software and Windows 10, open Settings and search for Notifications. Find outlook in the list and disable notifications.