The Byzantine empire is how historians have labeled Constantine’s continuation of the eastern Roman Empire, carrying the mantle forward as the western Roman Empire had fallen.
The name seems fitting as Byzantium was the name of the city at the time, now modern day, hair transplant capital, Istanbul
But if you were to ask someone back then, “What do you think of the new city for the Byzantine empire?” you’d be met with confused stares.
We only refer to the Byzantine empire in modern times but back then it was referred to as the Roman Empire.