1. The problem involves understanding the union of three open sets $U_1$, $U_2$, and $U_3$ in a topological space $X$ and the concept of faces of a simplex. 2. A face of a simplex is defined as a simplex formed by a subset of the vertices of the original simplex. 3. Each face has a vertex set that is a subset of the vertex set of the original simplex. 4. The union theorem in topology states that the union of any collection of open sets is open. 5. Since $U_1$, $U_2$, and $U_3$ are open sets in $X$, their union $U_1 \cup U_2 \cup U_3$ is also open in $X$. 6. The intersection of vertex sets of faces is nonempty if the faces share common vertices. 7. This means that the faces of a simplex overlap at vertices that are common to them. 8. Therefore, the union of these faces covers the simplex, and their intersections correspond to shared vertices. 9. In summary, the union of open sets $U_1$, $U_2$, and $U_3$ is open, and the faces of a simplex correspond to subsets of vertices with nonempty intersections representing shared vertices.