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We develop some of the foundations of affinoid pre-adic spaces withoutNoetherian or finiteness hypotheses. We give some explicit examples of non-adicaffinoid pre-adic spaces (including a locally perfectoid one). On the positiveside, we also show that if every affinoid subspace of an affinoid pre-adicspace is uniform, then the structure presheaf is a sheaf; note in particularthat we assume no finiteness hypotheses on our rings here. One can use ourresult to give a new proof that the spectrum of a perfectoid algebra is an adicspace.

Stably uniform affinoids are sheafy