The complement of the star at v is the union of all simplices not having v as a vertex

Let {T_β} _β be the simplices in a simplicial complex K not having v as a vertex. Let U be the star at v. Then Int T_β is in the complement of U and so is their union and closure of the union . By the previous post, it is equal to the union of the closures, i.e., the union L of {T_β} _β. So L is contained in the complement of U. To prove the converse, let {S _α} _α be the set of simplices having v as a vertex. If x is not in L then x is in the interior of a simplex S _α, so x is not contained in the complement of U. i.e. the complement of U is contained in L. QED