What is it that forces the Mail app to be open in order to receive a notification? Messages doesn't need to be open and those still come through. I agree, having to have Mail open to receive a notification is a big disappointment. Why can't Mail be pushed like on the iPhone?
The short answer:
Mail notifications are local notifications, not push. Most e-mail services don't do push. Without Mail running, your computer has no way to know that you have new mail.
The long answer:
There's a lot of confusion going around. I'll try and help clear things up. First, I want to point out one thing: This
IS how it works on the iPhone. Mail, being one of the stock apps, has some special abilities that 3rd party apps don't have. One of those is that the process to get new mail is always running (or at least running at the interval you have set for how often it should check new mail). So just like in Mountain Lion, Mail has to be running to give you notifications.
Now, why is it that Mail must be running to receive notifications, you ask? After all, Messages doesn't need to be running to receive iMessages.
The difference with iMessages is that it uses
push notifications. Push notifications are "pushed" from the remote server to the device that displays them, as opposed to
local notifications, where the Notification Center is simply told by an app running on the machine to display a notification.
So why can't Mail use push notifications then, you perhaps ask as a follow-up question?
Well, technically, it can. iCloud mail for example, uses push. But unlike iMessages, e-mail is an open world of services outside of just iCloud (Gmail, etc.), and Mail is designed to be open to work with all of them. But push is NOT standard in the sense that an e-mail service is not generally expected to support it (in fact, most don't). So rather than Gmail or whatever pushing a message to your computer telling you you have new mail, Mail.app actually queries Gmail (checks for new mail), and if there is any new mail, it tells Notification Center to show a notification. That's the important part:
the mail service doesn't push mail to your computer, Mail.app has to ask the mail service if there is new mail. If it's not running, it can't do that. Sure, they could make some background process that's always checking for new mail, whether Mail is open or not, but that wouldn't be good, and wouldn't change the fact that an additional process needs to be running to get the notifications.
Yes, Mail could probably allow push notifications for service like iCloud, that do support push e-mail, but that would be inconsistent and misleading, making people expect that to work for all their e-mail.
So it's not Apple's fault or a shortcoming in Mail or the Notification Center that require Mail to be running: it's just the way e-mail works.
The solution? Just leave mail running, but hide it (cmd-h, or right click the Dock icon and click "Hide") to make it invisible. Or close the mail window. Or whatever. Those of us who've been using GrowlMail up until ML have been doing it this way for years, it works just fine.
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I may not be understanding your post, but I do get push notifications when I get a new email on my iPhone, even when the Mail app is not running in the background. Can you clarify?
Mail is always running in the background. Being one of the stock apps, it has special privileges that other apps don't have.