Vertex set of an abstract simplicial complex/Definition

Definition

Let $\mathcal{S}$ be a abstract simplicial complex, we define the vertex set of $\mathcal{S}$, denoted as just $V$ or $V_\mathcal{S}$, as follows:

• $$V_\mathcal{S}:\eq\bigcup_{A\in\{B\in\mathcal{S}\ \vert\ \vert B\vert\eq 1 \} } A$$ - the union of all one-point sets in $\mathcal{S}$

Note: we do not usually distinguish between $v\in V_\mathcal{S}$ and $\{v\}\in\mathcal{S}$, they are notionally identified.