The reason people continue to refer to "Byzantium" (this was not the name they used to themselves) is that the Western European countries also made the claim to

The reason people continue to refer to "Byzantium" (this was not the name they used to themselves) is that the Western European countries also made the claim to be the heirs of the Roman Empire. So they couldn't accept the fact that "Byzantium" was the real Roman Empire (at least what remained of it).