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