Open and closed stars in a subdivision

For a vertex $v$ of a complex $T$ define its open star by ${\mathrm{st}}_T(v)=\{x\in \mathrm{\mid }T\mathrm{\mid }:x_v>0\}$. Define its closed star to be the subcomplex ${\mathrm{St}}_T(v)=\{\tau \in T:\tau \cup \{v\}\in T\}\mathrm{.}$ These definitions apply separately to $T=K$ and $T=\mathrm{sd}K$ from Barycentric subdivision of an abstract simplicial complex; in the latter, vertices are nonempty original faces. Coordinates always refer to $\mathrm{\mid }T\mathrm{\mid }$ as in The geometric realization of an abstract simplicial complex.

On each simplex the condition $x_v>0$ is open, so the open star is weakly open. The closed star is closed under faces; its realization meets every simplex in a finite union of faces and is weakly closed. If $x\in \mathrm{\mid }\tau \mathrm{\mid }$ and $\tau \cup \{v\}\in T$, then $(1-t)x+t{e}_v$ for $0<t\le 1$ is in the open star and converges to $x$ in that finite simplex. Hence the realization of the closed star is exactly the closure of the open star. Merely listing simplices containing $v$ would omit faces and would not define a subcomplex.

2.C, p.178, star paragraph and Lemma 2C.2.