I'm not looking for the definition of discrete topology given in textbooks; I'm wondering why the word 'discrete' was chosen.
I mean, the concept of discrete topology is built up from sets, which are built from objects--which are discrete. So, if we're looking for a word to differentiate power sets as topologies from other topologies, and we use the adjective 'discrete' to accomplish that differentiation because the power set is composed of discrete objects--then, by similar reasoning, couldn't we call all topologies on sets 'discrete'.
Because they're built from discrete objects and compositions, too. All of them. All topologies.
There must be some other reason we call discrete topologies 'discrete'. What is it?