In mathematics, given a universal set S , and a family of sets K over S , K is an abstract simplicial complex if the following is true:

: ∀ X ⊆ S , if X ∈ K , then ∀ Y ⊂ X it follows that Y ∈ K .

The elements of K are called abstract simplices. Furthermore, for X ∈ K , define the dimension to be dim(X) = | X | − 1 , and consequently define dim(K) = max{dim(X), X ∈ K} .

One-dimensional simplicial complexes are (simple) graph (mathematics)s.

: K(d) = {X ∈ K, dim(X) ≤ d}

is the d-skeleton of K . In particular, the skeleton (topology) is called the underlying graph.

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