How to prove that a simplicial complex is path-connected if connected?

Question

If K consists of finite simplices and connected, it seems intuitively clear that any two 0-simplices can be connected by a path whose image is a collection of 0-simplices and 1-simplices.

But I can't rigorously construct such a path... Could anyone help me with it?

Answer 1

Prove and use the following two lemmas:

Lemma. A connected and locally path-connected^$\dagger$ space is path-connected.

When a space is locally connected, the path components are open. So if $C$ is a path component, it's open, and its complement is a union of other path components, which are open so $C$ is closed too.

Lemma. A simplicial complex is locally path-connected.

A simplicial complex is locally homeomorphic to $\mathbb{R}^k \times [0,\infty)^{n-k}$.

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^$\dagger$ A space is locally path-connected when every point has a basis of path-connected neighborhoods.