•why did the fall of the roman empire provide an opportunity for the byzantine empire to rise?

Actually, the Byzantine Empire was the Eastern Empire of the Roman empire, When Emperor Diocletian came in the empire was split into 2 Western and Eastern Empire, as the Western fall to the Mongols the East was very lucky to Live on since Emperor Diocletian lived there but overall the answer would be It Increase since there were no Romans to destroy them or bother them.I hope this is the Answer if Not my Apologies.