A topological net structure and a topological graph

This paper describes a coherent topological structure from the discrete case to the continuum. The applied mathematical methods include a shortcut of a nonstandard analysis approach. A topological structure ( Z², τ₄) is developed which is adapted to the discrete situation. The structure is investigated for Jordan´s curve theorem and furthermore equipped with the required properties for applications like homogeneity and the equivalence between path-connected and topological connectivity. The compatibility of this topological structure to morphological operators is given