What empire lasted almost 1000 years after splitting from the western roman empire?

For the Romans it was the Roman Empire. This is how the Romans called it, they had no other terms.

Historians have invented the terms Eastern Roman Empire for the eastern part and Western Roman Empire for the western part. Byzantine Empire has been coined in reference to the eastern part of the empire after the fall of the west. The Word Byzantine is derived from Byzantium, the Greek city which was later turned into Constantinople. It had been chosen to highlight the fact that within just over a century after the fall of the west it assumed a Greek character. Greek replaced Latin as the official language of this empire in 620.