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Questions about maps into the powerset of a set (called set-valued maps, multivalued maps, or relations), corresponding concepts of continuity (like upper and lower semicontinuity), inclusion problems (like differential inclusions), maximal monotone maps, hyperspaces (families of subsets of a set, endowed with the Hausdorff distance or Vietoris topology), etc.
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Tangent cone of null sets
One can use your infinite-density example, but replace the outer lines with very sparse dotted lines:
$$S = (\{0\} \times \mathbb{R}) \cup \bigcup_{i=1}^\infty \{i^{-1},-i^{-1}\} \times \left[\bigcup_ …