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Topological simplicial complex

Definitions

For a simplicial complex $K=(V,\Sigma)$, the topological realisation $|K|$ is the quotient space formed by taking a copy $\Delta_\sigma$ of the standard simplex that corresponds to each simplex $\sigma$, and identifying the corresponding faces appropriately.

Properties

The topological realisation of K is the union of the insides of the simplices. These insides are disjoint.

An open set $U \in |K|$ is one where $\forall \sigma \in \Sigma: \Delta_{\sigma} \cap U$ is open in $\Delta_{\sigma}$

Given a vertex $v \in V$ the star of st$(v)$ is $\bigcup\{$inside$(\sigma): v \in \sigma \in \Sigma\}$. Stars are open sets.

If a simplicial complex is finite, then its topological realisation is compact.

If a simplicial complex has $n+1$ vertices, it is a subspace of the $n$-simplex, which is in turn a subspace of $\mathbb{R}^{n+1}$. From this, we can see that all finite simplicial complexes are compact and metrizable.

The topological realisation of any simplicial complex is Hausdorff.

Types

Connectedness: is connected if and only if is path-connected if and only if any two vertices in $K$ are joined by an edge path.

Examples

All simplicial subcomplexes are closed

all these pages adapted with probably insufficient credit from my university's lecture notes