Can a polygon be one dimensional? When looking up the definition of polygon, Wikipedia tells me: 

In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit.

Does this definition include sets of vertices like $\{(0,0),(5,0),(6,0),(0,0)\}$, which can be displayed in just one dimension?
 A: Your set of vertices satisfies all the terms of the definition, so it is technically a polygon by that definition.
Some would call it a degenerate polygon.
To disallow degenerate polygons, you will need to modify the definition, adding additional constraints.
EDIT: in the original post, I claimed that adding the condition that there exist at least non-collinear segments would remove the degenerate polygons. This is false: see comments.
A: These things are not universally defined. In some contexts   it would make sense to admit your example as a polygon, and in others it would not.
An example of the first context would be a discussion of a computer algorithm for detecting whether a point was interior to the polygon, or for calculating the area or the convex hull of a polygon.  One would expect the algorithm to work reasonably even for a degenerate polygon.
An example of the second context would be the study of plane tilings or tessellations, where degenerate polygons are uninteresting as tiles, or a discussion of the triangulation of manifolds into simplices, where the triangles are expressly required to be non-degenerate.
Typically (but not always) each author will state the particular definition or at least make a remark like “we exclude degenerate polygons”.
A: Seems to fit that definition.
However facets, being parts of polytopes such as polygons and polyhedra, shouldn't generally line up in the dimension below that of there dimension. For example edges shouldn't be collinear, faces shouldn't be coplanar, and points shouldn't be in the same place as each other. Note that star polygons are still fine.
It might be desirable to be able to switch between an "abstract polytope" like approach where polytopes are made of a set of such parts, for example a square has 4 edges, 4 vertices, and a face. And an approach where the polygon is made of the points that form its edge, a little like how a circle is defined. This causes ambiguity if adjacent edges are aloud to be in a line, or if points can be the same place as each other.
On a somewhat related note, tilings generally avoid 2 edge vertices.
