I am trying to prove that a simplicial complex is second countable if and only if locally finite. Since the simplicial complex lies in $\mathbb{R}^n$ (I am assuming geometric realization of simplicial complex) I was hoping I could take advantage of the fact that $\mathbb{R}^n$ is second countable in some sort of way.
Here is my attempt:
The definition of a closed set in $\vert K \vert$ (where $K$ is a simplicial complex) is that a set $A\in \vert K \vert$ is closed if and only if $\sigma \cap A$ is closed (in $\sigma$) for each simplex $\sigma$. Using this I (believe) I have formatted an equivalent definition for open sets. A set $O\in \vert K \vert$ is such that $C_{\vert K \vert}(O) \cap \sigma$ is closed for each $\sigma\in K$. Thus $C(C_{\vert K \vert}(O) \cap \sigma)$ is open which implies $C(C_{\vert K \vert}(O))\cup C(\sigma)$ is open which implies $O \cup C(\vert K \vert)\cup C(\sigma)$ is open. Since $C(\vert K \vert)$ is a subset of $C(\sigma)$ then I may write $O\cup C(\sigma)$ must be open for each $\sigma \in K$. Thus for a subset $O$ of $\vert K \vert$ to be open I require $O \cup C(\sigma)$ to be open for each $\sigma\in K$.
In order to show that $\vert K \vert$ is second countable I need to show that there is a countable basis for the open sets described above. Since some sets which are open in $\vert K \vert$ are not open in $\mathbb{R}^n$ I cannot finish this off with the fact that $\mathbb{R}^n$ is second countable (Not surprising since the topology on $\vert K \vert$ is finer than that of $\mathbb{R}^n$). This is where I am stuck, also another impending issue is that I have not used local finiteness...
In the other direction, suppose $\vert K \vert$ is second countable. Then there exists a countable basis of open sets $\mathcal{O}=\{O_{i}\}_{i=1}^{\infty}$ in $\vert K \vert$ such that every open set of $\vert K \vert$ can be written as the union of a subfamily of $\mathcal{O}$. Suppose for contradiction that $K$ is not locally finite, then there exists a vertex $v$ such that there exists an infinite number of simplices containing $v$. We shall label these simplices by $\sigma_{\alpha}$ where $\alpha\in J$ and $J$ is an indexing set. From here I am not sure what to do...
Would really appreciate a hint rather than a complete answer, but any help is welcome.