Vertex set of an abstract simplicial complex/Definition

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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 one-point 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.


  1. 1.0 1.1 Elements of Algebraic Topology - James R. Munkres