Grid cell topology

The grid cell topology is studied in digital topology as part of the theoretical basis for (low-level) algorithms in computer image analysis or computer graphics. 
The elements of the "n"-dimensional grid cell topology ("n" ≥ 1) are all "n"-dimensional grid cubes and their "k"-dimensional faces ( for 0 ≤ "k" ≤ "n"−1); between these a partial order "A" ≤ "B" is defined if "A" is a subset of "B" (and thus also dim("A") ≤ dim("B")). The grid cell topology is the Alexandrov topology (open sets are up-sets) with respect to this partial order. (See also poset topology.)
Alexandrov and Hopf first introduced the grid cell topology, for the two-dimensional case, within an exercise in their text "Topologie" I (1935).
A recursive method to obtain "n"-dimensional grid cells and an intuitive definition for 
grid cell manifolds can be found in Chen, 2004. It is related to digital manifolds.