When using Outlook.com, it can display email and event notifications even while closed. For this, the browser in which it is accessed should be open though.
Here is how to enable these desktop notification settings:
Login to Outlook.com and click on the gear icon from the top right.
Then click View all Outlook settings.
From the General > Notifications section, enable the option of desktop notifications. You can selectively choose these for email messages, events, or both.
For emails, this feature is available if you have the Focused Inbox enabled. These notifications are displayed even for messages sent to different Outlook aliases if present.
On enabling, there will be a confirmation pop-up in the browser to allow these notifications. Click on Allow.
A test notification is generated on doing so.
For Windows 10, all such notifications are shown in the Action Center. From now on, whenever new messages/events are available in Outlook.com, there will be desktop notifications for them even if Outlook is not open in the browser.