24th European Summer School in Logic,
Language and Information
Pointfree geometry and topology
An introductory course
Lecturer: dr Rafał Gruszczyński (Department of Logic, NCU in Toruń)


A brief description of the course
The main purpose of the course is to familiarize an audience with various pointfree
approaches to geometry, topology and reasoning about space. By a pointfree approach
I mean such in which points are not assumed to be primitive objects of a theory, but
are constructed from other entities (for example such that are more easily interpreted
in the surrounding world).
Prerequisites
 lattices and Boolean algebras
 basic geometry and topology
 metric spaces
Part I: Introduction
The main purpose of the introductory part of the course is to set the correct perspective on the problem.
First, it contains a refresher on fundamental notions from ordered sets theory and topology. Second, it
explains the meaning of the notion pointfree topology (geometry) we use in the whole course.
Third, we remind what an ordinary pointbased geometry is and how it can be done. Fourth, the origins
of pointfree approach to geometry are presented.
slides
Part II: Mereology
The third part is fully devoted to mereology, a mathematical theory
of part of relation, which will be further used as the main tool for
development of various pointfree theories of space. The most important
notion of this part is that of a mereological sum (mereological set,
collective set, fusion) of a group of objects. An axiomatization of mereology is given and
the connection with the theory of Boolean algebras is explained. All the necessary material to grasp
the ideas from this part of lectures can be found in (Pietruszczak, 2005).
slides
Part III: Basic assumptions of pointfree geometry
This part begins with an analysis of the Russellian perspective space from (Russell, 1914)
We present Russell's arguments for pointfree theory space which is motivated by properties of the
perspective space. Afterward we
move on to presentation of Stanisław Leśniewski view's on geometry. In the final stage we present the set of
postulate and properties which may be called basic methodological and ontological assumptions of
pointfree theories of space.
slides
Part IV: Grzegorczyk's system of topology
Part IV treats the system of pointfree system of topology as constructed by Andrzej Grzegorczyk in
(Grzegorczyk, 1960) with
further analysis and emendations from (Biacino et al., 1996) and (Gruszczyński et al., 2011).
In this part the following important notions are introduced and examined: of separation of regions,
connection of regions, nontangential inclusion of regions, quasiseparation structure
and separation structure. By means of these we introduced some special sets of regions which are
called prepoints. From these we construct points as filters generated by the former in
separation structures. Further it is demonstrated that the notion of the open set is captured by the
Grzegorczyk's theory and resulting topological space must always be Hausdorff.
slides
Part V: Tarski's geometry of solids
Tarski's system of pointfree geometry based on mereology was first presented in (Tarski, 1929) and
further slightly modified and fully developed in (Gruszczyński et al., 2008).
Tarski's idea was to enrich the axioms of mereology with one primitive
notion of ball and series of definitions and axioms that let him introduce him the notion of
point. Thanks to his simple but ingenious idea Tarski was also able to capture in his system the
ternary notion of equidistance of points, thanks to which (and due to Mario Pieri's discovery from)
he could talk about Euclidean geometry. This part of the lectures is devoted to presentation of aforementioned
Tarski's ideas.
slides
Part VI: Pointfree metric spaces and verisimilitude
theories
Pointfree metric spaces as presented in (Gerla, 1990) are structures that are similar to
metric spaces but such in which
the notion of point is replaced with that of a region and a distance function on points is
replaced with a distance between regions. An additional notion of the diameter
of a region is also introduced. After the presentation the basics of the theory of pointfree metric spaces
we show an application of the theory o the solution of the problem from the philosophy of science —
the Popperian idea of verisimilitude of scientific theories (Gerla, 2007).
slides
Bibliography
 L. Biacino, G. Gerla “Connection structures:
Grzegorczyk's and Whitehead's definitions of point”, Notre Dame Journal of Formal Logic,
vol. 37, no. 3 (1996), pp. 431–439
(full text PDF)
 T. de Laguna “Point, line and surface as set of solids”,
Journal of Philosophy, vol. 19 (1922), pp. 449–461
 G. Gerla “Pointless metric spaces”,
Journal of Symbolic Logic, vol. 55, no. 1 (1990),
pp. 207–219
(full text PDF)
 G. Gerla “Pointless geometries”,
Handbook of Incidence Geometry, edited by F. Buekenhout and W. Kantor, NorthHolland,
Amsterdam (1995), pp. 1015–1031
(full text PDF)
 G. Gerla “Pointfree geometry and verisimilitude of theories”,
Journal of Philosophical Logic, vol. 36, no. 6 (2007),
pp. 707–733
(full text PDF)
 R. Gruszczyński, A. Pietruszczak “Pieri's structures”,
Fundamenta Informaticae,
vol. 81, no. 1–3 (2007), pp. 139–154
 R. Gruszczyński, A. Pietruszczak “Full development of
Tarski's geometry of solids”,
The Bulletin of Symbolic Logic, vol. 14, no. 4 (2008),
pp. 481–540
(full text PS)
 R. Gruszczyński, A. Pietruszczak “Space, points and mereology. On foundations of
pointfree Euclidean geometry”, Logic and
Logical Philosophy, vol. 18, no. 2 (2009), pp. 145–188
(full text PDF)
 R. Gruszczyński, A. Pietruszczak “How to define mereological
(collective) set”,
Logic and Logical Philosophy, vol. 19, no. 4 (2010),
pp. 309–328
(full text PDF)
 R. Gruszczyński, A. Pietruszczak “On Grzegorczyk's system of pointfree
topology”, unpublished paper (2011)
 Handbook of spatial logics, edited by M. Aiello, I. PrattHartmann, J. van Benthem,
Springer (2007)
 A. Pietruszczak Metamereology (in Polish), Nicolaus Copernicus University Press, Toruń (2000)
 A. Pietruszczak “Pieces of mereology”, Logic and Logic Philosophy ,
vol. 14, no. 2 (2005), pp. 211–234
(full text PDF)
 B. Russell Our knowledge of external world, The Open Court Publishing Company, 1914.
 A. Tarski “Les fondements de la gèometriè de corps”,
pages 29–33 [in:] Księga pamiątkowa Pierwszego Polskiego Zjazdu Matematycznego,
supplement to Annales de la Societè Polonaise de Mathèmatique}, Kraków 1929.
English translation: “Fundations of the geometry of solid”,
pages 24–29 [in:] Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Oxford 1956.
