Topology is the mathematical study of sets of objects classified as open or closed and transformations of such sets and classifications. From this simple beginning, we can derive space, functional analysis, connectedness and compactness concepts, as well as tools useful in the study of quantum physics, psychology, computer science and more. Topology began with the study of the bridges of a northern city, literally, map-making. How is this act of map-making the act of sets of points and their transformations? Well, the points of an actual city are grouped into classifications like parks, counties, connected or separated areas, and then these points (taken from a 3-D globe) are transformed, or related to, the points of a 2-D sheet of paper. So map-making raises the general questions of this intellectual activity. Furthermore, certain transformations of a city, like road maps, preserve some information, like road connections, but lose other information, like elevation above sea-level. In general, we explore ideas like open, closed, connected, separate, information-preserving and information-sacrificing transformations. These are the very ideas of space, time, and measurement. It is therefore no surprise that quantum physics has made extensive use of topology. Furthermore, our minds are maps. We don’t actually experience the world as it is. For example, we experience light reflected from various objects mapped onto our occipital, or vision, cortex in the back of the head. Some information, like red-wavelength light absorption, is preserved in this mapping, and some, like x-ray radiation, is not. So it is no surprise that neuropsychology and neurology have made extensive use of topology. In his TED lectures, Steven Johnson, writer of the book Emergence and occasional participant in The Daily Show, illustrated how making a map of London was essential to understanding the source of cholera outbreaks, leading to great changes in public health and allowing the evolution of large cities that were livable. Topology is a root discipline that touches fundamentals of physics, mathematics, logic, psychology and social and political theory. Because topology is a fundamental exploration of transformations of sets of objects and the information (limit points) lost or preserved in that mapping, and our experience of life is a topological transformation of the world onto our mind, topology magically touches every discipline of human knowledge.
The beginning of topology is the definition of a topology, given a set U and collection of subsets T:
1) U and {} are in T
2) The union of any elements of T is in T
3) The intersection of any finite elements of T is in T
This collection is a topology.
For example, consider the set U = {1,2,3,4}
Here is a topology T= {1,2,3,4},{},{1,2},{3},{1,2,3}
Of course, things are much more interesting, with a infinite number of points like in Euclidean space. All sets of a topology are defined as “open”. I see and remember this by thinking of the objects in the open subsets as flowing freely into one another. The topological basis (a concept we can’t go into here) of Euclidean space is the collection of open space-balls. Any space may be constructed of overlapping balls of space of different size. Functions are seen as transformations of this topology. Questions of continuity and differentiability of functions in functional analysis also find a home in topology.
Anyone wishing to learn more should consult Topology by Munkres, or Topology by Hocking.
“I’m the map, I’m the map, I’m the map, I’m the map. I’m the map! . . . And which character would be singing that? . . . ” Brian Regan
Enough is Enough 