From TopCom, Volume 8, #1
This is a summary of a talk given at the conference on Subtle Technologies, Toronto, Canada, May 2001.It is not widely known that abstract pure mathematics has applications in daily life. There are many examples in mathematics but in this talk I wish to explain what is Topology and its applications. I'll use simple examples from day to day life to illustrate the concepts. Topology deals with the concept of nearness at various levels. This concept was first formulated by the Hungarian mathematician F. Riesz about 100 years ago in an address to the International Congress of Mathematicians in Rome (1908).
Level 1: Consider a typical family {Mother, Father, Son, Daughter}. We can say that a person is near the family if that person is blood related to the family. Of course, every member of the family is near the family and the family must first exist to talk about nearness! Grandparents, aunts, uncles, cousins, ... are persons near the family though they are not in the family. There are many other ways of having such nearness relations e.g. we can say that a person is near the family if the person helps the family in some way. In this definition the family doctor, the plumber, the mailman, ... are near the family. This concept is axiomatized with a few simple obvious conditions and we get the abstract concept of a topological space. This was first done by the Polish mathematician K. Kuratowski in 1922.
With a topological space, is associated another concept that of a continuous transformation. Suppose five years back Physician P was a family doctor of the family F . That is P was near the family F under the rule that P helps the family in some way. Today, after five years, both the Physician P and the family F have changed. If P is still near the family F , we say that it is a continuous relationship in day to day life and the same is said in mathematics. If for any reason, P ceases to be the family doctor of F , we say that the relationship is discontinuous. Thus a continuous transformation is one in which the nearness of a point to a set remains unchanged under that transformation.
To recapitulate, in TOPOLOGY, we have nearness relations between points and sets, together with continuous transformations which preserve these nearness relations.
Level 2: At this level we talk about nearness between two families, technically called a proximity space. This idea, already present in Riesz's work, was thoroughly studied by the Soviet mathematician V. Efremovic around 1940 and published in 1951. This idea was further developed by the Soviet mathematician Yu. Smirnov Again, this nearness between two families, can occur in several ways: (a) two families can be near because a daughter from one family has married a son from another, or (b) two families have a common friend (i.e., a person near both families), or (c) two families are interested in music and meet at a concert thus getting near each other. Moreover, in this subject we study transformations which preserve nearness between pairs of sets. These are called proximally continuous transformations. Level 3: Here we can talk of nearness of a number of families technically resulting in a uniform space discovered by the French mathematician A. Weil in 1937. An example is that of the families of persons who work for the same company or families. Perhaps the families get together for a picnic or a Christmas party. Again we have transformations which preserve this nearness of families and these are called near transformations or uniformly continuous transformations. In the most general sense this was studied by the German mathematician Horst Herrlich and the author in early 1970s.
You can see that the subject is international and mathematicians from all over the world have worked on this topic. We work on this topic because the problems are interesting, challenging, or beautiful! Sometimes problems come from other areas but there are many instances where applications are found later. Abstract topology has found applications in Theoretical computing, Quantum Mechanics, Relativity, Mathematical Economics, Optimization, Convex Analysis, Probability Theory, Theory of Capacities, Child Psychology, a model of our eyes, etc.
Date: January 21, 2003.