Q:

Definition and Example of Polytope Shelling.

I'm trying to get this definition of polytope shelling right through an example, but I'm afraid I might be misunderstanding it. 

By the definition and remark above, this "shelling" of a tetrahedron should work.

But doesn't we run into a problem when $F_2$ is added to the "shelling" after $F_1$? Since the shelling of the boundary complex of $F_2$ is 1-dimensional, but $F_2 \cap F_1$ is a point, I don't see how that would work.

A:

$F_2 \cap F_1$ isn't a point, it's an edge.
Then $F_3 \cap (F_1 \cup F_2)$ consists of two edges, and $F_4 \cap (F_1 \cup F_2 \cup F_3)$ consists of three edges.

