Mail notifications continue to appear even when mail app is closed
Notifications for mail are designed to arrive whether the main app runs or not.

You can disable notifications for mail entirely, though - Go to System Preferences > Notifications and make sure the apps have their Notification settings turned off. In the example below, the DaisyDisk alert style has been set to None:



You can also uncheck all the checkboxes for more granular control of when you see (and hear) notifications.