The Origins of Set-Theoretic Topology

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The Origins of Set-Theoretic Topology

Overview

Topology was a generalization of geometry that began to assume a separate identity in the nineteenth century. The study of set theory was first made a serious part of mathematics at the end of the nineteenth century. The two were combined in the twentieth century to the advantage of both, as set theory (which until then had seemed to have little to do with ordinary mathematics) found an application in topology and topology (which was capable of wandering off in many directions from geometry) could have a place of its own with the help of set theory. Many of the traditional areas of mathematics could be restated in general terms with the help of set-theoretic topology, which gave a new range of application to terms like "function."

Background

Geometry was one of the earliest branches of mathematics to assume identity as a separate discipline, largely thanks to the efforts of Euclid (fl. c. 300 b.c.) in his work the Elements. Euclid remained the source for all matters geometric well into the nineteenth century, although some by that time had come to worry about the specific form that the axioms (the assumptions from which Euclid started) took. In general, the material of the Elements covered topics like the areas of plane figures and the volumes of solids as well as the more abstract domains of geometry. Those in other branches of science could use the results of geometry even if they did not need to follow the proofs that Euclid supplied.

In the early nineteenth century several mathematicians began to investigate what would happen if some of the axioms of Euclidean geometry were dropped or altered. This non-Euclidean geometry did not seem to have any immediate application to the external world, but it could be connected with a variant of ordinary geometry called "projective" geometry. This study had attracted the attention of painters who were seeking to represent on a flat surface a three-dimensional image. Girard Desargues (1591-1661) had written an entire volume on projective geometry, which still seemed remote from the geometry of Euclid ordinarily taught in the classroom In the nineteenth century mathematicians like Arthur Cayley (1821-1895) in England recognized the possibility of connecting projective geometry with the rest of mathematics by treating it as an interpretation of non-Euclidean mathematics.

One of the central notions of Euclidean geometry was that of congruence of figures, which had to be the same size and shape. In projective geometry the idea of size dropped out of consideration and the notion of shape became central. For example, the Sun is known to be much bigger than a coin, but if the coin is held at an appropriate distance from the eye it will block out the sun exactly to the viewer. What is important for projective geometry is that the two objects could be made to appear the same size, a notion that was formalized in the work of Cayley and others.

German mathematician Felix Klein (1849-1925) used the work on projective and non-Euclidean geometries to produce a grand scheme for classifying all the various sorts of geometry that had begun to appear. In Euclidean geometry two figures could be considered the same if they had the same size and shape. In projective geometry it was enough if they had the same shape, even if they were not the same size. Still more general approaches to geometry could be introduced into this scheme, including topology, which took two figures to be the same if one could be turned into the other by bending or stretching but without tearing. (In technical terms this kind of transformation was known as a continuous deformation.)

The other ingredient that contributed to the creation of set-theoretic topology was the discipline of set theory, introduced into mathematics by Georg Cantor (1845-1918). Ideas about the infinite had been in existence in philosophical and popular scientific literature for many years, but there seemed to be paradoxes involved that would prevent their being used in mathematical applications. Cantor was able to define the central features of infinite sets in a way that would make sense of the paradoxes. Cantor had invented his set theory for use with the representation of functions, although there were other mathematicians at the time who felt quite uneasy about the use of the infinite in mathematics itself.

Impact

In Euclidean geometry straight lines and more complicated figures were made up of points, although the connection between the points and the line they made up was obscure. After all, the line was made up of an infinite number of points, and if a single point were omitted from the line, that did not seem to change the line itself. What was needed was precisely the idea of infinite set with which Cantor dealt. This provided the basis for the generalization of the terminology from ordinary geometry to the newer kinds of relationships and figures studied in topology. The word topology came from the Greek word topos, which referred to a place, and the word topography had been in existence for the making of maps. The objects with which topology had to deal were much more abstract, however, than the kind of maps put together in an atlas. For example, points in the new discipline of topology frequently did not represent just locations in a plane. The points could be used to represent functions and other mathematical objects.

Perhaps the most important contributor to the emergence of set-theoretic topology as a discipline was René Maurice Fréchet (1878-1973). Although his best-known book on abstract spaces did not appear until 1951, his ideas played a role in topology throughout the first half of the twentieth century. He recognized how to extend the ideas usually applied to the real numbers to the points that were generated from more complicated objects. Of these, perhaps the most crucial was that of distance, since the distance between two real numbers was easily defined as the absolute value of their difference. In more than one dimension, the Pythagorean theorem could be used to calculate the distance between points as the square root of the sum of the squares of the distances in the individual dimensions.

Fréchet observed that there were certain features about the distance function that were crucial to proceed to the rest of mathematics—for example, the triangle inequality held in ordinary geometry, which claimed that the sum of the lengths of two sides of any triangle was always greater than the length of the third side. Fréchet laid down that inequality as a requirement for the distance function when applied to points in the general spaces he had created. When this was combined with a couple of other requirements, a more general kind of geometry was possible, thanks to the additional abstractness of considering functions as points.

The work of Cantor allowed for even more general spaces than those for which a distance function was designed. German mathematician Felix Hausdorff (1868-1942) wrote a textbook on topology in 1914, perhaps the first to deal with the new subject. Cantor had defined a subset of a set to be a collection of objects contained in the set, so that (as an example) the even numbers are a subset of the whole numbers. In ordinary geometry one defined the objects near another object in terms of distance. For Hausdorff a neighborhood of a set could be defined in terms of subsets without even having a distance function with which to start. The basis for the choice of subsets that would define the neighborhood was the geometry of the real numbers, but the new definition could be given in an entirely abstract setting.

Hausdorff, Fréchet, and other contemporaries went to work trying to translate as much of the mathematics of standard geometry into a form applicable for the more general spaces of topology. In the language of the real numbers, an interval was the set of numbers between two given points. For topology this notion had to be replaced by that of a connected set, one that could not be separated into two sets by use of the neighborhoods that had been defined. One of the most important notions in calculus as performed on functions on the real numbers was that of continuity, which was often expressed in terms of being able to draw the graph of a function without having any gaps. In topology the notion of continuity was expressed in terms of open sets, a generalization of the open intervals from the ordinary real numbers. All the theorems of ordinary calculus had to remain true under the new definitions of topology, which encouraged even those taking more traditional mathematics as their domain to give the definitions a form more easily translated into the general setting.

Not all centers for mathematical research were equally quick to pick up set-theoretic topology as a subject worth investigation. After all, there had been plenty who objected to Cantor's introduction of set theory in the first place, and topology was not so thoroughly accepted as to carry much confidence. It is all the more striking that a relatively new mathematical culture like that of Poland (which had only become independent again after many years of foreign government in 1918) made a conscious effort to specialize in the areas of set-theory and topology. With the support of mathematicians elsewhere who were interested in the subject and of Polish mathematicians who had gone abroad, research into functional analysis—the application of set-theoretic topology to generalizations of problems from calculus—boomed. The periodical Fundamenta Mathematica may have been Polish in place of publication but its contributors were from all over the mathematical world.

Many members of the Polish school were forced either to go abroad during the Second World War or fell victim to the German invasion, but their results were disseminated through the centers for research elsewhere as well. Waclaw Sierpiński (1882-1969) was for many years editor of Fundamenta Mathematica and spent much of that time on set-theoretic topology. Of even greater importance was the work of Kazimierz Kuratowski (1896-1980), which proved to be even more important, as he used the ideas of Boolean algebra (the ideas of basic set theory familiar from before Cantor) to produce even more general spaces. In ordinary geometry one thinks of a boundary of a set as the points that define the shape of the figure. Then the closure of a set is the original set together with its boundary. Kuratowski took the idea of a closure operator more generally and defined the topology of an abstract space entirely independent of the notion of points. This proved to be more fertile than Fréchet's approach.

The area of set-theoretic topology blossomed in the hands of mathematicians from many countries. In addition to those mentioned above, Russia, England, and Hungary all were home to mathematicians who made contributions to set-theoretic topology. It remains an interesting historical example of a discipline that was appropriated by the mathematicians of one country (Poland) and made their specialty. One kind of abstract topological space is even called a Polish space in recognition of the central role played by the centers of research in Poland in promoting the subject.

The early part of the twentieth century saw the generalization of many branches of mathematics, much to the consternation of those who did not see the need to go beyond the traditional divisions. One of the lessons of the century has been that even problems stated in traditional terms may have a solution most easily found using weapons from a more abstract arsenal. Set-theoretic topology has supplied many of the tools involved in both stating new mathematical challenges and in solving old ones.

THOMAS DRUCKER