Why is the simplicial star of a vertex contractible? Let $S$ be a simplicial complex and $v$ is a vertex. Then the simplicial star of $v$ in $S$  is defined to be union of closed simplices containing $v$. Then the result is
The simplicial star of a vertex $v$ in a simplicial complex $S$ is contractible.
I can see the pictorial proof which will be to drag the points to the vertex along the simplices in which they belong.  Can anybody give a proof which is more precise by which I mean using the properties or constructions of simplicial complexes.  Thanks in advance.
 A: Every simplex which contains $v$ as a face is contractible. We contract each simplex via its own map. If each map agrees with every other map on the parts which are shared (for example, two adjacent tetrahedra have contractions which agree on the triangle which is the intersection of them), then we can apply the pasting lemma to create one contraction of the entire star.
We can ensure that this is the case because the intersection of two simplices in a simplicial complex is either empty or another simplex. Since a simplex is contractible, every map is homotopic to every other map (formally, the contraction is defined on $(Simplex)\times I$, which is also contractible). Thus when presented with two contractions $f_\alpha, f_\beta$ of different simplices, $S$ and $T$ with nonempty intersection, we homotope $f_\alpha|_{S\cap T}$ to $f_\beta |_{S\cap T}$ and then apply the homotopy extension property to render a new map $f'_\alpha$ which contracts S but now agrees with $f_\beta$ on $S\cap T$. Performing this process for every pair of maps which share part of their domain, we can create a contraction of the entire star.
