Apple’s “Mail” does not show when opened

The notifications that you receive for Mail within Notification Center are local notifications, not push notifications from some Apple server. As such, the program has to be running in order for the action that will trigger a notification to be able to do so.

Notification Center cannot check for new mails by itself, it has no in-built capabilities for polling/checking etc in order to create notifications, it merely receives them and displays them.

It is possible that in due course some email app will be made available that supports push notifications, but at present none do, including Apple's own. The fact that you get such notifications on iOS is more a feature that the mail client is always running in order to check for new mail, and create the local notifications accordingly. Leaving mail open all the time on your Mac isn't really an issue, it's hardly resource intensive, and if upon receiving a mail alert you wish to act upon it, then opening the alert will only open mail for you anyway.

The best way to summarise it is that NC just listens for alerts that other applications create, and displays them according to your preferences. It takes no part in the creation of alerts.

Your messages are not gone.

Apple Mail is not displaying the Exchange or G-mail inbox properly.

I tested this with 2 systems: 1 on Mavericks and 1 on Yosemite.

The Mavericks mail worked correctly and the Yosemite would stop displaying the inbox. I was able to perform operations (read, flag, delete, move) messages on the Mavericks system and the actions would correctly show on the Yosemite system once I re-synced the mail application.

Furthermore, I only needed to click on my iCloud-based account and then go back to my Exchange account for the inbox to be displayed properly. However, after about 30-45 seconds it would again stop displaying.

This looks like an Apple bug.

I hope this helps and that Apple can fix this annoying bug soon.