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.

## See also

## References

*Digital Geometry: Geometric Methods for Digital Image Analysis*, by Reinhard Klette and Azriel Rosenfeld, Morgan Kaufmann Pub, May 2004, (The Morgan Kaufmann Series in Computer Graphics) ISBN 1-55860-861-3*Topologie*I, by Paul Alexandroff and Heinz Hopf, Springer, Berlin, 1935, xiii + 636 pp.- Chen, L. (2004).
*Discrete Surfaces and Manifolds: A Theory of Digital-Discrete Geometry and Topology*. SP Computing. ISBN 0-9755122-1-8.

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