Abstract Simplicial Complex

In mathematics, given a universal set S, and K, a family of sets over S, K is an abstract simplicial complex if the following is true:
for each X \subseteq S, if X \in K then for each subset Y \subset X it follows that Y \in K.
The elements of K are called abstract simplices. Furthermore, for X \in K define the dimension dim X = |X| - 1 and consequently \dim K = \max \{\dim X, X \in K\}. One dimensional simplicial complexes are called (simple) graphs. Finally, the K^{(d)} = \{X \in K| \dim X \leq d\} is called the d-skeleton of K. In particular, the 1-skeleton is called the underlying graph.

See also

 

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