# simplicial complex definition

The simplicial complex is defined as:
A simplicial complex ${\mathcal {K}}$ is a set of simplices that satisfies the following conditions:

1. Any face of a simplex from ${\displaystyle {\mathcal {K}}}$ is also in ${\mathcal {K}}$.
2. 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?

• Of course one can. But having faces does make it easier to define a boundary operator .... – Lord Shark the Unknown Sep 25 '17 at 6:31

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 $\{[a],[a,b]\}$.
• Just because a set has an element that has an element doesn't mean the set has that second element. $\{1,\{1,2\}\}$ vs $\{1,2,\{1,2\}\}$. The first set only indirectly contains $2$. – Kyle Miller Sep 25 '17 at 17:32