How does this proposition imply that every compact region in the plane bounded by a regular polygon is a closed $2$-cell?

Question

From Intro to Topological Manifolds by Lee:

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After the proof, the author writes: "every compact region in the plane bounded by a regular polygon is a closed $2$-cell".

However, these regions:

although not convex, are compact and bounded by some polygon.

So, how does this proposition imply that every compact region in the plane bounded by a regular polygon is a closed $2$-cell?

Answer 1

Presumably, when Lee says "bounded by a regular polygon", he does not mean "contained within a regular polygon"; after all, a finite collection of points in the plane is also bounded in this sense, and it's certainly not a 2-cell.

Instead, I would take "region bounded by a regular polygon" to mean "the entire region enclosed by a regular polygon", which is convex. Lee seems to be using "polygon" to mean just the boundary, rather than this whole region.