Other Recursively Defined Classes of Complexes

In this section we mention some other classes of abstract simplicial complexes that are studied in Combinatorial Algebraic Topology and are defined in a recursive way. These families are less prominent than the nonevasive complexes, so we shall keep our presentation brief. 

Definition 13.28. A pure abstract simplicial complex X is called vertex-decomposable if it is empty or if there exists a vertex v of X such that both the link lkx v) and deletion dlx v) are vertex-decomposable. 

For example, the simplicial complex consisting of a single vertex is vertex-decomposable, since both the link and deletion of this vertex are empty. Furthermore, any simplex is vertex-decomposable. 

Definition 13.29. A pure abstract simplicial complex X is called con-structible if it is a simplex, including the empty simplex, or if it has con-structible subcomplexes Y and Z such that 

We finish by stating without proof the following important proposition. 

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