Pattern?

While Fabrizio was here I noticed a thread running through some of the math I am interested in: sets with the property that they intersect every *special* subset nontrivially. What *special* is differs in context. For example:

stationary sets:
intersect every closed unbounded set of ordinals (in some alpha)
piecewise syndetic sets:
closure (in the Stone-Cech compactification) intersects every two-sided ideal
simple sets:
intersect every recursively enumerable subset of integers