What is the intersection of the vertices of a face of a simplicial complex?

Question

I am currently reading "Subgroup graph methods for presentations of finitely generated groups and the contractibility of associated simplicial complexes" By Cora Welsch and I'm a bit stuck with Theorem 5.6.

She considers a subcomplex $U$ of the nerve complex $NC(G,H_{fi})$.

I dont understand why:

1. There always exists a finite set $\Sigma$ consisting of all maximal simplices of $U$?
2. What is the intersection of the vertices of an element $ \sigma \in \Sigma $? (She denotes it by $\cap \sigma$.)

Thanks in advance!