The difference between polyhedral complex and support of a polyhedral complex?

A polyhedral complex is a collection of polyhedra such that intersection of any two polyhedron is a face of of both the polyhedron or empty.

Support of a polyhedral complex is the set of all points in the polyhedral complex. So, what exactly is the difference between polyhedral complex and its support?

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Can you think of two different polyhedral complexes which have the same support? – Rahul Apr 1 '12 at 20:54
Well, I don't understand what exactly is the difference. If two polyhedral complexes have the same support, shouldn't they be the same polyhedral complex, according to the definition of polyhedral complex? – Mohan Apr 2 '12 at 7:40

For example, in two dimensions, you can overlay a polyhedral complex over the set $[0,1]\times[0,1]$ in many ways: as a single square-shaped polyhedron; as two isosceles right triangles; as an $n\times n$ grid of squares of side $\frac1n$; and so on. All of these are different polyhedral complexes, but they have the same support.