# Some topological properties of discrete surfaces

## Abstract

A basic property of a simple closed surface is the Jordan's property: the complement of the surface has two connected components. We call *backcomponent* any such component, and the union of a backcomponent and the surface is called the *closure* of this back-component. We introduce the notion of *strong surface* as a surface which satisfies a strong homotopy property: the closure of a back-component is strongly homotopic to that back-component. This means that we can homotopically remove any subset of a strong surface from the closure of a backcomponent. On the basis of some results on homotopy ([2]), and strong homotopy ([3], [4], [5]), we have proved that the simple closed 26-surfaces defined by Morgenthaler and Rosenfeld ([19]), and the simple closed 18-surfaces defined by Malgouyres ([15]) are both strong surfaces. Thus, strong surfaces appear as an interesting generalization of these two notions of a surface.

## Keywords

Closed Surface Strong Surface Simple Point Singular Surface Discrete Surface## References

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