# Faces of a simplex in a simplicial complex is in the simplicial complex

A simplicial complex $$K$$ is a set of simplexes $$\{\sigma\}$$ with the following conditions satisfied:

1. If $$\sigma' \leq \sigma \in K$$, then $$\sigma' \in K$$ ($$\sigma' \leq \sigma$$ means that $$\sigma'$$ is a face of the $$\sigma$$).

2. If $$\sigma, \sigma' \in K$$, then it is either $$\sigma \cap \sigma' = \emptyset$$ or $$\sigma \cap \sigma' \in K$$.

Given (b) below, it does not satisfy the condition 2. I want to know if it satisfies the condition 1 or not. It seems to me that it does satisfy the condition 1, since all of the faces of every simplex in the simplicial complex seems to be in the simplicial complex. Indeed, I cannot think of a set of simplexes where condition 1 is not satisfied while condition 2 is satisfied. Can someone help me by suggesting one?

(Image reference: Nakahara, Geometry, Topology, and Physics)

Take $$K = \{\{a, b, c\}\}$$. Condition 2 is trivially satisfied, since there is only a single element in $$K$$. However, condition 1 is not satisfied: For instance we have that $$\{a\} \subset \{a, b, c\}$$, but $$\{a\} \notin K$$.
Let $$s$$ be a k complex. Remove a (k-1) face from $$s$$, then the simplicial complex obtained satisfies condition 2 because the removed (k-1) simplex cannot be obtained by intersections of other faces and it doesn't satisfy condition 1.