Editing Simplicial complex (section)

== Closure, star, and link ==
<gallery class="center" widths="350" heights="112">
File:Simplicial complex closure.svg|Two {{color|#fc3|simplices}} and their {{color|#093|'''closure'''}}.
File:Simplicial complex star.svg|A {{color|#fc3|vertex}} and its {{color|#093|'''star'''}}.
File:Simplicial complex link.svg|A {{color|#fc3|vertex}} and its {{color|#093|'''link'''}}.
</gallery>

Let ''K'' be a simplicial complex and let ''S'' be a collection of simplices in ''K''.

The '''closure''' of ''S'' (denoted <math>\mathrm{Cl}\ S</math>) is the smallest simplicial subcomplex of ''K'' that contains each simplex in ''S''. <math>\mathrm{Cl}\ S</math> is obtained by repeatedly adding to ''S'' each face of every simplex in ''S''.

The '''[[Star (simplicial complex)|star]]''' of ''S'' (denoted <math>\mathrm{st}\ S</math>) is the union of the stars of each simplex in ''S''. For a single simplex ''s'', the star of ''s'' is the set of simplices in ''K'' that have ''s'' as a face. The star of ''S'' is generally not a simplicial complex itself, so some authors define the '''closed star''' of S (denoted <math>\mathrm{St}\ S</math>) as <math>\mathrm{Cl}\ \mathrm{st}\ S</math> the closure of the star of S.

The '''[[Link (geometry)|link]]''' of ''S'' (denoted <math>\mathrm{Lk}\ S</math>) equals <math>\mathrm{Cl}\big(\mathrm{st}(S)\big) \setminus \mathrm{st}\big(\mathrm{Cl}(S)\big)</math>. It is the closed star of ''S'' minus the stars of all faces of ''S''.