Byzantium has an interesting history.

It was once an ancient Greek city of classical antiquity, founded in the 7th century BC. In 330 AD, the site had a new name. Constantinople, capital of the Roman Empire. Constantinople would be renamed in 1930 to Istanbul. It is one of the oldest consistent capital cities on Earth. Maybe even the oldest continuously inhabited and used capital city in human history. Inherited one could say across multiple civilizations and Empires for hundreds of years.

'Inheritor' has no scientific definition in history. It's really more of a device than anything. A way of conveying meaning. An inheritor is someone who takes up a culture, a place, a political model, or a system of power. In that regard, yeah. Lots of historians have termed the Ottomans inheritors of Rome.

The Ottomans conquered the city of Constantinople in 1453. It was the death of what could be called the longest continuous Empire in human history, because the Byzantines didn't call themselves the Byzantines. They called themselves Romans. Their empire was the Roman Empire, inheritor of the legacy of Rome. In fact, before the divide of the Roman Empire into West and East, the city could be called Nova Roma: Second Rome. It was a major Roman power center before the Empire split, and while the Western World frequently remembers the fall of the Western Empire, it tends to sort of forget or sideline that the Eastern Empire continued on for nearly another thousand years.

Why do we -as in you and me, us modern folks - call it the Byzantine Empire?

Politics.

No really. In the West, we usually think of Rome from the perspective of the Latin Western tradition from which we descend. For us, the proper inheritor of 'Rome' was the Holy Roman Empire. This gets tangled up in a lot of things.

The Roman Empire split into a Western and Easter Empire in 285, with the capital of the West in Rome and the capital of the East in Constantinople. The Western Empire weakened and fell 4th and 5th centuries. But the Eastern Empire kept chugging along strong.

Religious disputes get involved. In the west, the Christian Church began centralizing power around the Patriarch of Rome, the Pope. The Patriarch of Constantinople generally thought this was somewhat silly and disagreed with the whole idea. The 'inheritance' of Rome became culturally and politically contentious as part of a centuries long and convoluted power struggle between various men crowned 'King of the Romans', the Patriarch of Rome, aka the Pope, the Patriarch of Constantinople, and the Eastern Roman Emperors, who called themselves Roman Emperors. After the Great Schism divided the Catholic and Eastern Orthodox Church this only got more contentious.

There's honestly way too much history here for me to encapsulate, I might as well write a peer reviewed article. Long story short, in the Latin west we tend to think of the Holy Roman Emperors as the proper inheritors of Rome and generally think of the Eastern Roman Emperors as those other guys who weren't really that Roman.

But the Byzantines never called themselves the Byzantines. That was something the Latin West did starting in the 15th century. And frankly speaking, the Byzantines had a lot more in common with the Rome of antiquity than any of the Holy Roman Empires, though that's treating the Byzantines as a monolith that never changed, which isn't entirely true.

There is today an increasing push among historians of late Antiquity and the Middle Ages to reexamine the Byzantines, their history, cultural significance, and their legacy. In a lot of ways, our view of them is heavily painted by 1000s of years of old politics and some scholars argue we do not have an accurate picture of the Empire. We tend to treat the Byzantines as splitting off and becoming their own thing (usually around the time of Constantine or Justinian), but that's something of a enforced cultural habit. Not something that actually reflects the empire or how the people who lived their viewed themselves or there cultural legacy and affiliation.