Indeed, and even after the fall of Constantinople there were various Byzantine Rump Staes in places like Trebizond. There is some historical chicanery in the classic Western European telling of the story of the 'Byzantine Empire'. No such country ever existed as such - the name was applied to the Eastern Roman Empire by German historians after the fall of Constantinople. While it still existed, the Eastern Roman Empire was referred to as the Roman Empire (or Roman Kingdom or simply Romania).