View source for Vertex scheme of an abstract simplicial complex/Definition

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{{Requires work|grade=E|msg=There's a lot of work required here. Like what does it mean for "vertices to span a simplex" and so forth? Also the notation for the definition of {{M|\mathcal{K} }} leaves a lot to be desired! It's implied {{M|n}} varies but it is ''not'' in the spirit of this site!}}
==Definition==
</noinclude>Let {{M|K}} be a [[simplicial complex]] and let {{M|V_K}} be the {{link|vertex set|simplicial complex}} of {{M|K}} (not to be confused with the [[vertex set of an abstract simplicial complex]]), then we may define {{M|\mathcal{K} }} - an [[abstract simplicial complex]] - as follows{{rEOATJRM}}:
* {{MM|\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\} }}<sup>'''Warning:'''</sup><ref group="Note">{{m|n\in\mathbb{N}_0}} here so {{M|n}} may be zero, we are expressing our interest in only those ''finite'' members of {{M|\mathcal{P}(V_K)}} here, and that are non-empty. 
* {{XXX|This needs to be rewritten!}}</ref> - that is to say {{M|\mathcal{K} }} is the set containing all collections of vertices such that the vertices span a simplex in {{M|K}}
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==Notes==
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==References==
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{{Definition|Homology Theory|Simplicial Homology|Algebraic Topology}}
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