Same

I like to use two unrelated concepts in mathematics to form my username. In this case, I picked base in topology and domain of discourse in logic. I feel like imposing some form of topological structure on the domain of discourse seems like a nice idea.

Realistically, every topology has a base, however the idea of a baseless space sounds pretty cool in my head (basically means you cannot uniquely generate continuous function just from its behavior on the base, in this case it is equivalent to the only base being the entire space). Hence the name "baseless discourse".

As for it happens to make sense in day-to-day English, it is a happy accident :)