# How to prove that the intersection of two polyhedrons is still a polyhedron?

I've met a problem in M.A.Armstrong's Basic Topology.

If $K$ and $L$ are complexes in $\mathbb{E}^n$, show that $\vert K\vert\cap\vert L\vert$ is a polyhedron.

where $\vert K\vert$ and $\vert L\vert$ are the polyhedron of $K$ and $L$.

I think it's not hard to imagine this statement. But I can't find a formal proof for this.

EDIT： The definition of a polyhedron in Basic Topology is:

... the union of the simplexes which make up a particular complex is a subset of a euclidean space, and can therefore be made into a topological space by giving it the subspace topology. A complex $K$, when regarded in this way as a topological space, is called a polyhedron and written $\vert K\vert$.

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What if the two polyhedra touch each other's faces? –  Guess who it is. May 10 '11 at 2:58
What does |X| is the polyhedron of X mean? –  M.B. May 10 '11 at 3:38
@M.B. : I've edited the post for explanation. –  Roun May 10 '11 at 4:40