Let $K$ be a simplicial complex and if $\sigma$ is a simplex of $K$.

How to prove that the following sets are simplicial complexes?

1) The boundary of $K$: $\partial(K)=\{\mbox{proper faces which belong to all the simplexes of K}\}$

2) The closure of $\sigma$: $\mathrm{Cl}(\sigma)=\{\mbox{faces of }\sigma\}$

Can you help me please? Thank you!!!

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What definition do you have of simplicial complexes? – Dedalus Nov 24 '12 at 19:36
I have to prove this definition: A simplicial complex K is a finite collection of simplices in some R^n satisfying: 1. If σ ∈ K, then all faces of σ belong to K. 2. If σ, τ ∈ K, then either σ ∩ τ = ∅ or σ ∩ τ is a common face of σ and τ. – lauren Nov 25 '12 at 8:27