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I want to verify that the r-skeleton of a geometric complex is a geometric complex.This question was arised related with the topic geometric complexes and polyhedra.Can you help me? Besides this problem,I want to get an answer for the question that "How many faces does an n-simplex have?" and how can I prove it? Please answer me soon.

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What properties are you having trouble verifying? –  Qiaochu Yuan Apr 30 '11 at 18:23

1 Answer 1

A geometric model for an $n$-simplex is the set of all points $\vec{x}\in\mathbb R^{n+1}$ such that $x_i\geq 0$ and $\sum_{i=1}^{n+1}x_i=1$. The way you get to a codimension $1$ face is to let one of the coordinates go to $0$. There are $n+1$ coordinates, so there are $n+1$ faces. This matches our knowledge for low dimensions. A $2$-simplex is a filled-in triangle and has three faces (edges). A $3$-simplex is a tetrahedron which has $4$ faces.

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