How did the Byzantine empire react after the fall of Rome

They probably cheered.

First of all, when did the Western Roman Empire actually fall? The question is typical of our modern thinking and implies that the Western Empire fell in some kind of cataclysmic event (kinda like the island of Atlantis sinking beneath the ocean waves in a single day….). And just like that Atlantis event is a myth, so is the ‘Fall’ of the Roman Empire a myth.

the urdan working class