So now the question is... Does this shape have 3 vertices and 2 sides, 3 vertices and 3 sides, or 3 vertices and infinity+2 sides?
Under the definition given in that link I gave ("an open, nonempty subset of the boundary of a simple closed curve that is continuously differentiable"), it would actually have one edge and one vertex.
I don't have any experience in brown townysis, but I wonder if it would be possible to construct a figure with infinite edges with some variation of
this.