By definition, a simplicial complex K is a set of simplices that satisfies the following 2 conditions:
1. Any face of a simplex from K is also in .
2. The intersection of any two simplices σ1 and σ2 in K is a face of both σ1 and σ2.
I understand what 2 is trying to say, but I'm not so sure about 1. Isn't this condition automatic?
Eg - is our simplicial complex is simply a triangle (the 2-simplex) then it's faces are the vertices (0-simplices), the edges (1-simplices) & the triangle itself. & surely all of these faces are automatically included in the simplicial complex. ie - condition 1 holds. I can't think geometrically of a case when this would not be so. :-s