Preview is closed in Mavericks when last window is closed

You need to disable Preview's automatic termination.

defaults write com.apple.Preview NSDisableAutomaticTermination -bool TRUE && killall -HUP Finder

This is due to Lion's automatic termination which was introduced so that apps which have no windows open and are not performing a task can be quit to save system resources. Generally it's good to leave on, but if you really want to disable it, use the above command.

You can disable automatic termination in all apps by setting it in the global preferences.

defaults write -g NSDisableAutomaticTermination -bool TRUE && killall -HUP Finder

When you open a file in Preview.app, then close the last window, Preview appears to shutdown too. It disappears from the Dock.

However, if you check your Activity Monitor.app, Preview actually isn't shutdown. It still is listed in the CPU and Memory tab of Activity Monitor. For me, Preview is taking up 60MB of memory and it appears to be closed. It's not in my Dock or in the Menu at the top of screen.

If you open Preview up again. Then close it using cmd-Q or select "Quit Preview" from the menu, it will actually shutdown. It disappears from the CPU and Memory tabs in Activity Monitor.

You don't need to disable automatic termination because it is not automatically terminated.