Closure - Star - Link

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

• closure of 𝑆 (denoted 𝐶𝑙( 𝑆)) is the smallest simplicial subcomplex of 𝐾 that contains each simplex in 𝑆. 𝐶𝑙(𝑆) is obtained by repeatedly adding to 𝑆 each face of every simplex in 𝑆
• star of 𝑆 (denoted 𝑆𝑡(𝑆)) 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 having 𝑠 as a face. (Note that the star of 𝑆 is generally not a simplicial complex itself)
• link of S (denoted 𝐿𝑘(𝑆)) equals 𝐶𝑙(𝑆𝑡( 𝑆)) − 𝑆𝑡( 𝐶𝑙( 𝑆)). It is the closed star of 𝑆 minus the stars of all faces of 𝑆