= Finite-hole neighborhoods of a planar compact set
{title2=$K\subset\operatorname{int}N\subset N\subset V$}
Given a <compact> planar set $K$ and an <open> neighborhood $V$, a finite square-grid construction gives a <compact> polygonal neighborhood inside $V$ with finitely many complementary components. Remove thin polygonal corridors disjoint from $K$ joining holes that lie in the same component of $\mathbb C\setminus K$, and joining holes in its unbounded component to the exterior. Corridors can also include a prescribed point from each of finitely many specified bounded components. There remains a neighborhood whose bounded holes lie in distinct bounded components of $\mathbb C\setminus K$, each containing its chosen point. Standard small polygonal rounding gives a finite union of planar surfaces with <boundary>. The index map from its first integral <homology group> to one integer per bounded hole is an isomorphism. Triangulate a surrounding disc: a one-cycle bounds a two-chain whose coefficients are constant in each complementary component. Zero hole coefficients give a filling supported inside the neighborhood; the face chain of each hole supplies a dual cycle. The cycles can be sums of boundary curves in different components. This uses finite triangulation, not the general <Jordan curve theorem>.
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