Simplicial Complex

Properties

1. Simplicial Complex is path connected.

proof

A simplicial complex is locally path-connected. A simplicial complex is locally homeomorphic to $R^k\times {\left[0,\infty \right)}^{n-k}$. A connected and Locally path-connected space is path-connected. When a space is locally connected, the path components are open. So if C𝐶 is a path component, it’s open, and its complement is a union of other path components, which are open so C𝐶 is closed too. https://math.stackexchange.com/questions/1018403/how-to-prove-that-a-simplicial-complex-is-path-connected-if-connected

Example

Kakimizu Complex