One of the problems that some people have experienced after upgrading to OS X Mavericks is the inability to read new e-mail messages in Apple's included Mail e-mail client. While the program runs fine in most respects, for some e-mail providers the program appears to check for messages only when Mail is first opened, and then no longer does so until Mail is closed and re-opened.
This bug may have you frustratingly quitting and re-opening the program frequently to check for your e-mail. While Apple has not offered any details regarding the bug, it has acknowledged the issue in a new support document that gives a couple of workarounds.
In short, while the bug persists, those affected can force Mail to check for new messages without resorting to quitting and relaunching the program.
The first option is to choose "Take All Accounts Offline" from the Mailbox menu, and then select "Get All New Mail" (immediately below the first option). This will bring the accounts back online and have them check for new messages immediately.
Rather than using the Mailbox menu, Apple also offers a quicker means of accomplishing this. Just right-click Mail's toolbar and choose the option to customize it. Then drag the "Take All Accounts Offline" button to the toolbar, and ensure that the "Get Mail" button is there (it is present by default). When done, you can click these two buttons in sequence to perform the same action.
Another approach to doing this is via user hot-key shortcuts. While the option to take accounts offline does not have a hot key by default, you can assign one in the "Shortcuts" section of the Keyboard system preferences. In this section, select "App Shortcuts" in the list of options, then click the plus button to add a shortcut. Next, select "Mail" from the drop-down menu and type "Take All Accounts Offline" in the Menu Title box. Now click the shortcut box and press your desired shortcut key (I recommend Shift-Command-period since you can then simply tap "N" to invoke the second shortcut).
When finished, you can then bring Mail to the front and type the shortcuts in succession to check your mail messages.
Hopefully this bug will be addressed in the next update to OS X Mavericks, but until then this option should allow you to check for new messages without having to close and relaunch the program.