The following exercise is drawn from Ch.24 of Fulton's "Algebraic Topology: A First Course."
Exercise 24.37 (a) and (b)
(a) If $U = \{U_v: v \in V\}$ is a finite collection of open sets whose union is a space $X$, define a simplicial complex, called the nerve of $U$ and denoted $N(U)$, by taking $V$ to be the set of vertices, and defining the simplices to be the subsets $S$ of $U$ such that the intersection of the $U_v$ for $v \in S$ is nonempty. Show that, in fact, $N(U)$ is a simplicial complex.
(b) In turn, if $K$ is any simplicial complex, and $v$ is a vertex of $K$, define an open set $St(v)$ in $|K|$, called the star of $v$, to be the union of the "interiors" of the simplices that contain $v$, i.e., $St(v)$ is the complement in $|K|$ of the union of those $|\sigma|$ for which $\sigma$ does not contain $v$. Show that the open sets $\{St(v): v\in V \}$ form an open covering of $|K|$, and that the nerve of this covering is the same as $K$.
For (a), I want to verify that $N(U)$ satisfies the definition of abstract simplicial complex, because I would think the desired claim follows from there. I think that verifying this will not be too hard, but I don't know if this strategy is appropriate, and wanted to see if anyone visiting would approach the problem differently.
As for (b), I don't have as well-defined a strategy in mind, and wanted to see how anyone visiting might approach the problem.