Atheism is the lack of belief in a god(s), it doesn't require an underlying belief. If you don't have a belief in a god then that automatically makes you an atheist.
Both theist and athiest can be further broken down into subsets if you bring claims of knowledge into the question.
I'm an agnostic atheist. I don't claim there are no gods. I would however claim the god as described in the Bible doesn't exist as the description is deeply flawed and contradictory.