Abstract simplicial complex
 See Books:Combinatorial Algebraic Topology  Dmitry Kozlov as this allows the emptyset as a simplex and the emptyset as a complex! Alec (talk) 11:27, 19 February 2017 (UTC)
Contents
Definition
Let [ilmath]\mathcal{S} [/ilmath] be a collection of sets, it is an abstract simplicial complex, or ASC for short, if^{[1]}:
 [ilmath]\forall A\in\mathcal{S}[A\neq\emptyset][/ilmath]
 [ilmath]\forall A\in\mathcal{S}[\vert A\vert\in\mathbb{N}_0][/ilmath]  where [ilmath]\vert\cdot\vert[/ilmath] denotes cardinality of a set
 [ilmath]\forall A\in\mathcal{S}\forall B\in\mathcal{P}(A)[B\neq\emptyset\implies B\in\mathcal{S}][/ilmath]^{[Note 1]}
Any [ilmath]A\in\mathcal{S} [/ilmath] is called a simplex of [ilmath]\mathcal{S} [/ilmath], and any [ilmath]B\in(\mathcal{P}(A)\{\emptyset\})[/ilmath] is called a face of [ilmath]A[/ilmath]
 Caveat:We may allow both the empty set to be an asc and we may also allow the empty set to be a simplex  as per Books:Combinatorial Algebraic Topology  Dmitry Kozlov
Terminology
 We make the following definitions regarding dimension of an abstract simplicial complex:
 For any simplex, [ilmath]A\in\mathcal{S} [/ilmath], we define: [ilmath]\text{Dim}(A):\eq\vert A\vert1[/ilmath]  the dimension of [ilmath]A[/ilmath] is one less than the number of items in the simplex considered as a set
 We define the dimension of the abstract simplicial complex itself as follows: [math]\text{Dim}(\mathcal{S}):\eq\mathop{\text{Sup} }_{A\in\mathcal{S} }\Big(\text{Dim}(A)\Big)[/math]
Related terminology
 Warning: do not confuse vertex scheme with the vertex set!
Vertex Set
Let [ilmath]\mathcal{S} [/ilmath] be a abstract simplicial complex, we define the vertex set of [ilmath]\mathcal{S} [/ilmath], denoted as just [ilmath]V[/ilmath] or [ilmath]V_\mathcal{S} [/ilmath], as follows^{[1]}:
 [math]V_\mathcal{S}:\eq\bigcup_{A\in\{B\in\mathcal{S}\ \vert\ \vert B\vert\eq 1 \} } A[/math]  the union of all onepoint sets in [ilmath]\mathcal{S} [/ilmath]
Note: we do not usually distinguish between [ilmath]v\in V_\mathcal{S} [/ilmath] and [ilmath]\{v\}\in\mathcal{S} [/ilmath]^{[1]}, they are notionally identified.
Vertex Scheme
The vertex scheme of a simplicial complex, [ilmath]K[/ilmath], is an abstract simplicial complex.
Definition
Let [ilmath]K[/ilmath] be a simplicial complex and let [ilmath]V_K[/ilmath] be the vertex set of [ilmath]K[/ilmath] (not to be confused with the vertex set of an abstract simplicial complex), then we may define [ilmath]\mathcal{K} [/ilmath]  an abstract simplicial complex  as follows^{[1]}:
 [math]\mathcal{K}:\eq\left\{\{a_0,\ldots,a_n\}\in \mathcal{P}(V_K)\ \big\vert\ \text{Span}(a_0,\ldots,a_n)\in K\right\} [/math]^{Warning:}^{[Note 2]}  that is to say [ilmath]\mathcal{K} [/ilmath] is the set containing all collections of vertices such that the vertices span a simplex in [ilmath]K[/ilmath]
See next
See also
Notes
 ↑ Perhaps better written as:
 [ilmath]\forall A\in\mathcal{S}\forall B\in(\mathcal{P}(A)\{\emptyset\})[B\in\mathcal{S}][/ilmath]
 ↑ [ilmath]n\in\mathbb{N}_0[/ilmath] here so [ilmath]n[/ilmath] may be zero, we are expressing our interest in only those finite members of [ilmath]\mathcal{P}(V_K)[/ilmath] here, and that are nonempty.
 TODO: This needs to be rewritten!

References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} Elements of Algebraic Topology  James R. Munkres