Simplicial complex

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Definition

A simplicial complex, $K$, in $\mathbb{R}^N$ is a collection of simplices, $K$, such that:

1. $\forall \sigma\in K\forall\tau\in$$\text{Faces}(\sigma)$$[\tau\in K]$
2. $\forall \sigma,\tau\in K[\sigma\cap\tau\neq\emptyset\implies(\sigma\cap\tau\in\text{Faces}(\sigma)\wedge\sigma\cap\tau\in\text{Faces}(\tau))]$
   • TODO: "The intersection of any two simplices is a face in each of them" is what he says, $\emptyset$ being a face would tidy this up slightly but I still think it is not a face!

Underlying set & topology

We use $\vert K\vert$ to denote the "underlying set" of $K$:

• $\vert K\vert:\eq\bigcup_{\sigma\in K}\sigma$ - as expected

To make $\vert K\vert$ into a topological space we require a topology, say $\mathcal{J}$ (so $(\vert K\vert,\mathcal{ J })$ is a topological space)

• $\mathcal{J}:\eq\left\{U\in\mathcal{P}(\vert K\vert)\ \vert\ \forall\sigma\in K[\sigma\cap U\text{ open in }\sigma]\right\}$ - recall $\mathcal{J}$ is the set of open sets of the topological space.
  • Equivalently: $\mathcal{J}:\eq\left\{U\in\mathcal{P}(\vert K\vert)\ \vert\ \forall\sigma\in K\exists V\in\mathcal{K}[\sigma\cap U\eq \sigma\cap V]\right\}$ where $\mathcal{K}$ is the topology of $\mathbb{R}^N$ - the usual topology from the Euclidean metric
    • Recall a simplex has the subspace topology for its topology.
    • TODO: Confirm a simplex has the subspace topology!
• We can also work with closed sets:
  • $A\in\mathcal{P}(\vert K\vert)$ is closed if and only if $\forall\sigma\in K[\sigma\cap A\text{ is closed in }\sigma]$

Terminology

• The underlying set, $\vert K\vert$ is sometimes called the polytope of $K$
  • A space that is the polytope of a simplicial complex may be called a polyhedron - but some topologists reserve this for the polytope of a finite simplicial complex

TODO: We are undecided on this

Comments

• The topology of $\vert K\vert$ may be finer than the topology $\vert K\vert$ would inherit as a subspace of $\mathbb{R}^N$. We form the following claim:

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The claim Munkres makes is:

• Suppose $A$ is closed in $\vert K\vert$ considered as a topological subspace of $\mathbb{R}^N$ then $A$ is closed in in $\vert K\vert$ with its topology as defined above.
  • i.e. $\text{closed in subspace}\implies\text{closed in space}$

What does this mean for open sets?

Equivalent definitions

• A collection of simplices, $K$, is a simplicial complex iff $\forall\sigma\in K\forall \tau\in\text{Faces}(\sigma)[\tau\in K]$ and $\forall\sigma,\tau\in K[\sigma\neq\tau\implies \text{Int}(\sigma)\cap\text{Int}(\tau)\eq\emptyset]$

Properties

• A map from a simplicial complex to a space is continuous if and only if the map restricted to each simplex in the complex is continuous also
  • Barycentric coordinate with respect to a point of a simplicial complex
• A simplicial complex is a Hausdorff space
• If a simplicial complex is finite then it is compact
• If a subset of a simplicial complex is compact then that subset is a subset of a finite subcomplex of the complex

TODO: There's more and clean up!

See also

• Simplicial subcomplex (Subcomplex (simplex) redirects there) - usual sub construction as encountered everywhere
• Simplicial p-skeleton
• Vertex (simplicial complex)
• Abstracit simplicial complex
• Simplex