= Ordinarily nowhere-dense sets need not be completely Ramsey
The cone $[E]^\omega$ on the even <integers> is nonempty and <Ellentuck topology> open, while it is closed and <nowhere dense> in the <ordinary topology on infinite subsets>. Choose a <finite-symmetric-difference parity colouring> on that cone and take one colour class. The class is still ordinarily <nowhere dense> because its <closure> lies in the cone, but every stem-free refinement inside the cone meets both colours. Hence it is not <completely Ramsey>.
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