= Gap-doubling closed family of infinite subsets
{title2=$a_{n+1}>2a_n$}
The family $D=\{\{a_1<a_2<\cdots\}:a_{n+1}>2a_n\text{ for all }n\}$ is ordinarily closed and <nowhere dense>. A failed gap is witnessed by a finite prefix, and any cylinder can be refined by adjoining consecutive large integers that fail the gap. Nevertheless $D$ intersects every infinite-subset cone in continuum many sets: a binary tree chooses two sufficiently large next points inside its reservoir at each step.
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