Topological foundations of design and control of multidimensional discrete systems

Local and global connectedness, arcwise connectedness, path connectedness, and order relation are among the main structural components of complicated neural networks, communications networks, and other multidimensional discrete systems. All such systems are finitely connected. They can be considered as subspaces of some integer product spaces, Alexandroff spaces, primitively derived spaces, or other subclass of primitively path connected spaces. Connectivity, local connectivity, and order relation in such spaces are examined here. Finite connectedness and lexicographic union of ordered topological spaces are especially important because of their widespread applications.<>