In a lot of my history classes Ancient Greece is always referred to as the cradle of western europe, but when we refer to byzantium it is always called eastern europe. Wouldnt Byzantium be considered the cultural successor of Graeco-Roman culture rather than HRE/Germanic tribes or the Gauls?
So to clarifiy my question is why isnt western europe classified as the germanic and gaullic tribes in antiquity and HRE and France in medieval, and eastern europe as Graeco-Roman in antiquity and Byzantium in medieval.
Am I missing something here in the way it was explained to me?
The terms 'eastern' and 'western' are not only relative to what area you are in, but also overgeneralization of a more complex reality. It depends what you consider the eastern/western dichotomy to be based on: culture, language, ethnicities, politics, religions etc.
That being said, we often consider Ancient Greece to be part of the quote and quote "Western World" due to the legacies Greece has had on our political system (democracy), as well as our philosophy.
"The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato" (Alfred North Whitehead)
This quote is a gross over-exaggeration, but it does show us the extent to which the greek philosophical tradition has influenced modern western thought.
Since ancient greek culture has had such a vast impact western society (linguistically, philosophically, politically, culturally, artistically etc.), we consider it to be the ancestor of our contemporary culture, more so than let's say Germanic tribes.
When we consider the Byzantine Empire to be 'Eastern' it is mainly to differentiate it from its Western counterpart, the Western Roman Empire.
We should see the cultural influence of one civilization on another as something fluid across areas. It's not like Germanic tribes directly and lead to medieval France and the Holy Roman Empire, and it's not like Ancient Greece single handedly lead to the Byzantine Empire. Instead, Ancient Greece has had a cultural legacy on both, whilst other cultures in Western Europe had an influence on Modern Western Europe, and other eastern cultures had an influence on the modern east.
Imagine it as a genealogical tree, with a common ancestor for both, but also with other separate and respective ancestors.
You might also want to check out The Making of the West: Peoples and Cultures by Lynn Hunt. It's a great book and it'll help you understand the fluid and smooth connections between different what we call the Ancient Western World and the Modern Western World.
It might help if we knew at what level and where you're taking the history classes.