Applications of General Topology by Som Naimpally

Topology Atlas Document # topd-48

Applications of General Topology

Som Naimpally

From TopCom, Volume 8, #1

Other document formats (3 pages): DVI; PostScript; PDF

General Topology (GT), as a distinct branch of mathematics, took shape in the middle of the twentieth century with the publication of the widely used books by Bourbaki and Kelley. With numerous applications to Functional Analysis, Theoretical Physics (Quantum Mechanics), Theory of Games, etc. the subject became prestigious. In both national and international conferences General Topologists gave invited addresses. Then slowly the subject began to be ignored by the Mathematical Establishment. The prevailing view is that GT is dead. At present, the situation has deteriorated to such an extent that General Topologists find themselves in a ghetto - see numerous articles by Henriksen in the Mathematics Intelligencer and TopCom at the website of Topology Atlas. For quite some time, several journals have stopped accepting papers in GT and graduate students are discouraged from specializing in it. In this note an attempt is made to show the usefulness of GT that is not widely known.

1. Quasi-uniform Spaces

In 1965 when Murdeshwar and I were working on Quasi-uniform Spaces, a prominent mathematician, working in probability told me, "Kelley's book contains all that is needed and further research in GT is not going to be of much value." From about a dozen publications in 1965, the subject of q-u spaces has blossomed into about 500 publications mainly due to the excellent work done by Fletcher, Lindgren, Hunsaker, Kunzi, Brummer et al. A detailed account is written by Kunzi ([3]). It is interesting to note that recently this subject has found applications in Theoretical Computer Science. A speaker at the Prague Topological Symposium in 1991 said, "those working in quasi-uniform spaces were not aware that they were proving results in Theoretical Computing".

2. Generalized Metric Spaces

The solution of the metrization problem by Bing, Nagata and Smirnov, led to a surge in the study of Generalized Metric Spaces (GMSs). A GMS is a topological space which is not metrizable but satisfies some condition fulfilled by metric spaces. Some of the examples are Stratifiable spaces, Developable spaces, Spaces with G_d-diagonals. A semi-metric space (X, d) is one in which d satisfies all the conditions of a metric space except the triangle inequality and the topology is related to the semimetric d by the condition:

x is in the closure of A iff d(x, A) = inf{d(x, a) : a in A} = 0.

A developable space is a semimetric space in which the semimetric d is USC and arose naturally in the metrization problem. So developable spaces were recognized to be an important class of topological spaces. They lie midway between general topological spaces and metrizable spaces. A Moore space is a regular developable space and many distinguished mathematicians worked on the famous problem: Is a normal Moore space metrizable? However, in a general semimetric space, spheres need not be open. With such unusual properties satisfied by GMSs, there is a belief among some that research on GMSs is too esoteric to be of any value except to get a tenure/promotion! It was a pleasant surprise to see that these spaces find applications in General Relativity ([2]). For a unified approach to many GMSs see [7] where it is shown that they can be characterized in terms of real valued functions.

3. Mathematical Morphology

([8, Preface to Vol. I])

Mathematical morphology was born in 1964 when G. Matheron was asked to investigate the relationships between the geometry of porous media and their permeabilities, and when at the same time, I was asked to quantify the petrography of iron ores, in order to predict their milling properties. This initial period (19641968) has resulted in a first body of theoretical notions (Hit or Miss transformations, openings and closings, Boolean models), and also in the first prototype of the texture analyser. It was also the time of the creation of the Centre de Morphologie Mathematique on the campus of the Paris School of Mines at Fontainebleau (France). Above all, the new group had found its own style, made of a symbiosis between theoretical research, applications and design of devices.

Some of the section or chapter headings: What good is Topology?, Review of Topology, Hausdorff metric, Topological structure for the closed sets in R², Physical consequences of the Hit or Miss topology, Topological properties of the Hit or Miss transformations, Connectivity criteria, Homotopic thinnings and thickenings, Connected components Analysis, Dilations on topological spaces, usc dilations, Metrics linked to connectivity, Umbrae and semi-continuity, Lipshitz functions in the space L₂, From continuity to digitalization.

4. The topology of the brain and visual perception

Ophthalmologists have found that when the sight is restored to a person who is born blind, the person has a topological vision for a certain initial period. During this period, the person cannot distinguish between a circle and a square. The person has to learn over a period to delineate between various closed curves. Using this as a basis, Zeeman has constructed a topological model of the brain and visual perception ([9]).

5. Topological Psychology

([4, 5]) This is a controversial topic in which mathematicians have varied opinions on it. Kurt Lewin was born in Mogilno, Germany in 1980 and along with Psychology, he studied Topology that was being developed during the early part of the last century. He was a child psychologist and developed his theories using topology, which others in the field found "a queer sort of mathematics". Lewin resisted formalizing his theory into a definite form so as not to make it rigid and inflexible. He introduced concepts casually and gradually developed them the through experimentation and observation. Lewin came to the USA in 1932 and became quite well known in Psychology and there is a considerable literature available on the internet. http://wise.fau.edu/~bmellor/courses/mat1933/faculty1.html Lewin is well known in Gestalt Psychology and there is a Kurt Lewin Institutein the Netherlands.

6. Mathematical Economics

([1]) A commodity space X is a metrizable space which is locally compact and second countable. A utility function f is a continuous real-valued partial function on X i.e. the domain p(f) of f is a closed subset of X. In mathematical economics, there is a need to put a suitable topology on the function space of utility functions. There is very little research available on topologies on the spaces of partial functions which arise naturally as inverse functions in elementary mathematics, as solutions of ODEs etc. For a reasonably complete bibliography see [6].

References

[1] K. Back, Concepts of similarity for utility functions, J. Math. Econ. 15 (1986), 129-142.
[2] R. Z. Domiaty and O. Laback, Semimetric spaces in General relativity (On Hawking-King-McCarthy's path topology), Russian Math. Surveys 35; 3 (1980), 57-69.
[3] H-P. Kunzi, Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of Asymmetric Topology, History of General Topology, Volume 3, Kluwer Academic Publishers, (2002), 853-968.
[4] R. W. Leeper, Lewin's topological and vector psychology: a digest and a critique, (1943). [5] K. Lewin, Principles of Topological Psychology, (1930) McGraw-Hill, NY. [6] S. A. Naimpally, A new uniform convergence for partial functions, Acta. Math. Hungarica 88 (1-2) (2000), 45-52. [7] S. A. Naimpally and C. M. Pareek, Generalized metric spaces via annihilators, Q and A in Gen. Top. 9 (1991), 203-226. [8] J. Serra, Image Analysis and Mathematical Morphology, Volume I (1982), Volume II (1992), Academic Press.
[9] E. C. Zeeman, The topology of the brain and visual perception, Topology of 3-manifolds and related topics, (1962), 240-256. Prentice-Hall.

Date: January 21, 2003.