What are some Byzantine achievements?
This is a great question.
First, Byzantium preserved the culture of the Roman empire (Roman and Greek learning). When the Roman Empire in the West collapsed in 476, Byzantium ensured transmission for future generations. This point alone is one of the greatest achievements of Byzantium. In connection to this, the Roman law code was transmitted. Emperor Justinian's work on the law code was prodigious, to say the least. When we add the fact that many Western countries used these law codes as the basis of their law codes, the significance of this act becomes enhanced.
Second, Byzantium also revived Europe through the Renaissance. When Constantinople was sacked in 1453, many of the intellectuals in Constantinople moved to the continent and they spurred on greater learning. Without this impulse toward intellectual fervor and creativity, the Renaissance may not have happened.
Unpacking the implication of these two points will yield great insights.