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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?

enter image description here

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

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2 Answers 2

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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 $\{a\} \notin K$.

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Figure (b) satisfies condition 1.

For your query, you can construct such an example by simple methods -

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.

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