View source for Abstract simplicial complex

{{Stub page|grade=E|msg=Unimportant page at the current time as it will not come up in my exam! However I think it is important! Notes for improvement:
* See [[Books:Combinatorial Algebraic Topology - Dmitry Kozlov]] as this allows the emptyset as a simplex and the emptyset as a complex! [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 11:27, 19 February 2017 (UTC)}}
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==Definition==
Let {{M|\mathcal{S} }} be a collection of sets, it is an ''abstract simplicial complex'', or ''ASC'' for short, if{{rEOATJRM}}:
# {{M|\forall A\in\mathcal{S}[A\neq\emptyset]}}
# {{M|\forall A\in\mathcal{S}[\vert A\vert\in\mathbb{N}_0]}} - where {{M|\vert\cdot\vert}} denotes [[cardinality of a set]]
# {{M|\forall A\in\mathcal{S}\forall B\in\mathcal{P}(A)[B\neq\emptyset\implies B\in\mathcal{S}]}}<ref group="Note">Perhaps better written as:
* {{M|\forall A\in\mathcal{S}\forall B\in(\mathcal{P}(A)-\{\emptyset\})[B\in\mathcal{S}]}} 
This is an exercise in [[equivalent ways of expressing a sentence in FOL]], and should be easy to see</ref>
Any {{M|A\in\mathcal{S} }} is called a ''[[simplex]]'' of {{M|\mathcal{S} }}, and any {{M|B\in(\mathcal{P}(A)-\{\emptyset\})}} is called a {{link|face|simplex, abstract simplicial complex}} of {{M|A}}
* {{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 {{link|simplex|abstract simplicial complex}}, {{M|A\in\mathcal{S} }}, we define: {{M|\text{Dim}(A):\eq\vert A\vert-1}} - the dimension of {{M|A}} 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: {{MM|\text{Dim}(\mathcal{S}):\eq\mathop{\text{Sup} }_{A\in\mathcal{S} }\Big(\text{Dim}(A)\Big)}}
==Related terminology==
: '''Warning: ''' do not confuse {{link|vertex scheme|abstract simplicial complex}} with the {{link|vertex set|abstract simplicial complex}}!
===[[Vertex set (abstract simplicial complex)|Vertex Set]]===
{{:Vertex set of an abstract simplicial complex/Definition}}
===[[Vertex scheme (abstract simplicial complex)|Vertex Scheme]]===
The ''vertex scheme'' of a [[simplicial complex]], {{M|K}}, is an ''abstract simplicial complex''. 
====Definition====
{{:Vertex scheme of an abstract simplicial complex/Definition}}

==See next==
* {{link|isomorphism|abstract simplicial complexes}}
==See also==
* [[Simplicial complex]]
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Homology Theory|Simplicial Homology|Algebraic Topology}}