Mereotopology begins in philosophy with theories articulated by A. N. Whitehead in several books and articles he published between 1916 and 1929, drawing in part on the mereogeometry of De Laguna (1922).
The first to have proposed the idea of a point-free definition of the concept of topological space in mathematics was Karl Menger in his book Dimensionstheorie (1928) -- see also his (1940).
[1] The theory of Whitehead's 1929 Process and Reality augmented the part-whole relation with topological notions such as contiguity and connection.
Despite Whitehead's acumen as a mathematician, his theories were insufficiently formal, even flawed.
By showing how Whitehead's theories could be fully formalized and repaired, Clarke (1981, 1985) founded contemporary mereotopology.
[2] The theories of Clarke and Whitehead are discussed in Simons (1987: 2.10.2), and Lucas (2000: ch.
The entry Whitehead's point-free geometry includes two contemporary treatments of Whitehead's theories, due to Giangiacomo Gerla, each different from the theory set out in the next section.
Although mereotopology is a mathematical theory, we owe its subsequent development to logicians and theoretical computer scientists.
More advanced treatments of mereotopology include Cohn and Varzi (2003) and, for the mathematically sophisticated, Roeper (1997).
Barry Smith,[3] Anthony Cohn, Achille Varzi and their co-authors have shown that mereotopology can be useful in formal ontology and computer science, by allowing the formalization of relations such as contact, connection, boundaries, interiors, holes, and so on.
Mereotopology has been applied also as a tool for qualitative spatial-temporal reasoning, with constraint calculi such as the Region Connection Calculus (RCC).
It provides the starting point for the theory of fiat boundaries developed by Smith and Varzi,[4] which grew out of the attempt to distinguish formally between Mereotopology is being applied by Salustri in the domain of digital manufacturing (Salustri, 2002) and by Smith and Varzi to the formalization of basic notions of ecology and environmental biology (Smith and Varzi, 1999,[7] 2002[8]).
Casati and Varzi (1999: ch.4) set out a variety of mereotopological theories in a consistent notation.
Casati and Varzi do not say if the models of GEMTC include any conventional topological spaces.
Casati and Varzi prefer limiting the ontology to physical objects, but others freely employ mereotopology to reason about geometric figures and events, and to solve problems posed by research in machine intelligence.
Lower case letters from the end of the alphabet denote variables ranging over the domain; letters from the start of the alphabet are names of arbitrary individuals.
Enclosure, notated xKy, is the single primitive relation of the theories in Whitehead (1919, 1920), the starting point of mereotopology.
Let parthood be the defining primitive binary relation of the underlying mereology, and let the atomic formula Pxy denote that "x is part of y".
Call the resulting minimalist mereological theory M. If x is part of y, we postulate that y encloses x:
If this were not the case, topology would merely be a model of mereology (in which "overlap" is always either primitive or defined).
Replacing the M in MT with the standard extensional mereology GEM results in the theory GEMT.
The axioms of GEM assure that this sum exists if φ(x) is a first-order formula.
Hence the dual of i, the topological closure operator c, can be defined in terms of i, and Kuratowski's axioms for c are theorems.
Adding C5-7 to GEMT results in Casati and Varzi's preferred mereotopological theory, GEMTC.
Note that the primitive and defined predicates of MT alone suffice for this definition.
The predicate SC enables formalizing the necessary condition given in Whitehead's Process and Reality for the mereological sum of two individuals to exist: they must be connected.
Given some mereotopology X, adding C8 to X results in what Casati and Varzi call the Whiteheadian extension of X, denoted WX.
If the underlying mereology is atomless and weaker than GEM, the axiom that assures the absence of atoms (P9 in Casati and Varzi 1999) may be replaced by C9, which postulates that no individual has a topological boundary:
When the domain consists of geometric figures, the boundaries can be points, curves, and surfaces.
What boundaries could mean, given other ontologies, is not an easy matter and is discussed in Casati and Varzi (1999: ch.