Void abstract simplicial complex

Some words about degeneracies occurring in this context are in order. In Definition 2.1 we have allowed the empty collection of sets. The corresponding abstract simplicial complex is called the void abstract simplicial complex, and is denoted by 0, or by. The void abstract simplicial complex is the only one that does not contain the empty set as one of its simplices. 

For any abstract simplicial complex A, there exists a unique simplicial map from the void abstract simplicial complex to A. Additionally, if A is not the void abstract simplicial complex, then there exists a unique simplicial map from the empty abstract simplicial complex to A. 

In the degenerate cases, the deletion of the vertex v from the abstract simplicial complex 0, v will give the empty simplex, whereas the deletion of the empty set from any abstract simplicial complex will give the void simplex. Furthermore, if 5 is a set of simplices of A, then we define the deletion of S by setting... 

For example, in a hollow tetrahedron, the abstract simplicial complex A consisting of all subsets of 1,2,3,4 except for the set 1,2,3,4 itself, a link of an edge consists of the two vertices that do not belong to that edge. Note also that the void simplex can never be the link of anything, since any link contains an empty set. 

The join of an arbitrary abstract simplicial complex with the void simplex is equal to the void simplex. 

Very often, for the sake of brevity, in case no confusion arises, we shall talk about the topological properties of the abstract simplicial complex A, always having in mind the properties of the topological space. 4. We note that the geometric realizations of the void and of the empty complexes are both empty sets. 

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