Proposition 5.1.1
If \(Q\) is a polyhedron, then the collection of faces \(\Phi(Q)\) is a polyhedral complex. Moreover, the collection of bounded faces
$$
\Phi^{\mathrm{bnd}}(Q)
:=
\{\,F \in \Phi(Q) : F \text{ is bounded} \}
$$
is also a polyhedral complex.
Proof
Let \(Q\) be a polyhedron. If \(F\in\Phi(Q)\) and \(F'\) is a face of \(F\), then by Proposition 3.3.1, \(F'\) is also a face of \(Q\). Hence \(F'\in\Phi(Q)\). Therefore, the containment property holds. Now let \(F,G\in\Phi(Q)\). By Proposition 2.3(ii), the intersection \(F\cap G\) is a face of both \(F\) and \(G\). In particular, \(F\cap G\) is a face of \(Q\), so \(F\cap G\in\Phi(Q)\). Therefore, the intersection property also holds. Thus, \(\Phi(Q)\) is a polyhedral complex. Finally, every face of a bounded faced is also bounded so the second claim follows. \(\blacksquare\)