---
tag: 456E0RCQR1
kind: concept
name: Digital topology
msc: [68U03, 68U10]
sources:
  - url: https://en.wikipedia.org/wiki/Digital_topology
same_as:
  wikidata: Q5276205
  wikipedia: Digital topology
---

## Summary

Digital topology studies notions such as connectedness, boundaries and curves for sets of points of the integer lattice $\Z^2$ or $\Z^3$.
Such sets model the pixels or voxels of digital images.
Its basic tools are adjacency relations between lattice points and the paths and connected components they define.
One also uses genuine topologies on $\Z^n$, such as the Khalimsky topology.
These notions underlie image processing tasks such as counting objects, tracing borders and thinning shapes.

## Explanations

### L1 human

A black-and-white picture on a screen is a set of black squares on a grid.
To say that a shape in the picture is "in one piece" or "has a hole", one must decide when two squares touch.
They may touch only along an edge, or also at a corner.
For example, two black squares that meet only at a corner form one piece under the corner rule but two pieces under the edge rule.
Digital topology makes such choices precise and checks that familiar facts, such as "a closed curve separates the inside from the outside", still hold.