The simplicial complex is defined as:
A simplicial complex ${\mathcal {K}}$ is a set of simplices that satisfies the following conditions:
- Any face of a simplex from ${\displaystyle {\mathcal {K}}}$ is also in ${\mathcal {K}}$.
- The intersection of any two simplices ${\displaystyle \sigma _{1},\sigma _{2}\in {\mathcal {K}}}$ is either ${\displaystyle \emptyset }$ or a face of both ${\displaystyle \sigma _{1}}$ and ${\displaystyle \sigma _{2}}$.
My question is why the (1) condition is required in this definition. Is it possible to have a set of simplices not containing faces of simplices?
A simplex has faces, but that only means a set containing that simplex indirectly contains the faces. The first condition is saying that the set actually contains those faces. Consider $\{[a],[b],[a,b]\}$ vs $\{[a],[a,b]\}$.