Again, it doesn't save battery to close apps (unless an app gets stuck in a loop or something). Background apps are frozen in memory when they are not in the foreground after a brief period where they are allowed to finish any processes. They are eventually closed by the system if the memory is needed for something else.
They only use battery in the background if you are actively using specific background tasks (like playing music or navigating, etc.). In addition, apps can periodically perform background updates, but those happen whether they are open or closed.