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

Definition

Let $K$ be a simplicial complex and let $V_K$ be the vertex set of $K$ (not to be confused with the vertex set of an abstract simplicial complex), then we may define $\mathcal{K}$ - an abstract simplicial complex - as follows:

• $$\mathcal{K}:\eq\left\{\{a_0,\ldots,a_n\}\in \mathcal{P}(V_K)\ \big\vert\ \text{Span}(a_0,\ldots,a_n)\in K\right\}$$ ^Warning: - that is to say $\mathcal{K}$ is the set containing all collections of vertices such that the vertices span a simplex in $K$

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