Vertex scheme of an abstract simplicial complex
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Not important as I wont be examined but I think it is very important to the subject! See Abstract simplicial complex as it has the same note and is why this page was created
Contents
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 1]}  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
 Every abstract simplicial complex is isomorphic to the vertex scheme of some simplicial complex
 Two simplicial complexes are linearly isomorphic if and only if their vertex schemes are isomorphic as abstract simplicial complexes
Notes
 ↑ [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!
