In mathematics, an abstract cell complex is an abstract set with
Alexandrov topology In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is open; in Alexandrov topologies the finite rest ...
in which a non-negative integer number called
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
is assigned to each point. The complex is called “abstract” since its points, which are called “cells”, are not subsets of a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
as is the case in Euclidean and
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
es. Abstract cell complexes play an important role in
image analysis
Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading bar coded tags or as sophi ...
and
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
.
History
The idea of abstract cell complexes (also named abstract cellular complexes) relates to
J. Listing (1862) and
E. Steinitz (1908). Also A.W Tucker (1933), K. Reidemeister (1938), P.S. Aleksandrov (1956) as well as R. Klette and A. Rosenfeld (2004) have described abstract cell complexes. E. Steinitz has defined an abstract cell complex as
where ''E'' is an abstract set, ''B'' is an asymmetric, irreflexive and transitive binary relation called the bounding relation among the elements of ''E'' and ''dim'' is a function assigning a non-negative integer to each element of ''E'' in such a way that if
, then