I'm being a little pedantic, but this is a little misleading:
> After the Roman Empire fell, its knowledge, writings, and customs were somewhat preserved in Western Europe, but their real preservation and translation occurred in the Byzantine Empire
This is, of course, because the 'Byzantine' Empire was the Roman Empire, and continued to operate as such for almost a millennia after its loss of territories in the West, some of which it regained and lost again during that time.
> and Islamic world.
Many, many texts were lost and destroyed during the very violent Muslim conquests. It wasn't until much later that mostly ex-Roman and ex-Persian subjects began reviving learning from what had survived.
> hen in the 15th century the lifestyle, innovations, and preserved writings from these places and others as far as China gained traction in Europe again.
An important factor in this was the threatened, and then eventual, destruction of the Roman capital, Constantinople. Learned scholars fled as refugees to the west, bringing their books with them. Without the Turks destroying what was left of the Roman Empire, there might have been no Renaissance.