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This article introduces planar ribbons, Vergili ribbon complexes and ribbonnerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topologicalspaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closureof a pair of nesting, non-concentric filled cycles that includes the boundarybut does not include the interior of the inner cycle. Each planar ribbon hasits own distinctive shape determined by its outer and inner boundaries and theinterior within its boundaries. A Vergili ribbon complex (briefly, ribboncomplex) in a CW space is a non-void collection of countable planar ribbons. Aribbon nerve is a nonvoid collection of planar ribbons (members of a ribboncomplex) that have nonempty intersection. A planar CW space is a non-voidcollection of cells (vertexes, edges and filled triangles) that may or may notbe attached to other and which satisfy Alexandroff-Hopf-Whitehead containmentand intersection conditions. In the context of CW spaces, planar ribbons,ribbon complexes and ribbon nerves are characterized by Betti numbers derivedfrom standard Betti numbers $\mathcal{B}_0$ (cell count), $\mathcal{B}_1$(cycle count) and $\mathcal{B}_2$ (hole count), namely, $\mathcal{B}_{rb}$ and$\mathcal{B}_{rbNrv}$ introduced in this paper. Results are given forcollections of ribbons and ribbon nerves in planar CW spaces equipped with anapproximate descriptive proximity, division of the plane into three boundedregions by a ribbon and Brouwer fixed points on ribbons. In addition, thehomotopy types of ribbons and ribbon nerves are introduced.

Ribbon Complexes & their Approximate Descriptive Proximities. Ribbon & Vortex Nerves, Betti Numbers and Planar Divisions