Reference: Elementary Concept of Topology by Paul Alexandroff (Translated by Alan E. Farley). Dover Publications, Inc. NY.
Prof. David Hilbert’s views via the preface to this little booklet: (June 1962, Gottingen).
Few branches of geometry have developed so rapidly and successfully in recent times as topology, and rarely has an initially unpromising branch of a theory turned out to be of such fundamental importance for such a great range of completely different fields as topology. Indeed, today in nearly all branches of analysis and in its far reaching applications, topological methods are used and topological questions asked.
Such a wide range of applications naturally requires that the conceptual structure be of such precision that the common core of the supeficially different questions may be recognized. It is not surprising that such an analysis of fundamental geometrical concepts must rob them to a large extent of their immediate intuitiveness —- so much the more, when in the application to other fields, as in the geometry of our surrounding space, an extension to arbitrary dimensions becomes necessary.
While I have attempted in my Anschauliche Geometrie to consider spatial perception, here it will be shown how many of these concepts may be extended and sharpened and thus, how the foundation may be given for a new, self-contained theory of a much extended concept of space. Nevertheless, the fact that again and again vital intuition has been the driving force, even in the case of all of these theories, forms a glowing example of the harmony between intuition and thought.
Thus, the following book is to be greeted as a welcome complement to my Anschauliche Geometrie on the side of topological systematization; may it win new friends for the science of geometry.
I. The specific attraction and in a large part the significance of topology lies in the fact that its most important questions and theorems have an immediate intuitive content and thus teach us in a direct way about space, which appears as the place in which continuous processes occur. As confirmation of this view, I would like to begin by adding a few examples to the many known ones:
i) One need only think of the simplest fixed-point theorem or of the well-known topological properties of closed surfaces such as are described, for instance, in Hilbert and Cohn-Vossen’s Anschauliche Geometrie, chapter 6. Published in English under the title Geometry and the Imagination.
2) The Jordan Curve Theorem:
A simple closed curve (that is, the topological image of a circle) lying in the plane divides the plane into precisely two regions and forms their common boundary.
3. The question which naturally arises now is: What can one say about a closed Jordan curve theorem in three-dimensional space?
The decomposition of the plane by this closed curve amounts to the fact that there are pairs of points which have the property that every polygonal path which connects them (or, which is “bounded” by them) necessarily has points in common with the curve. Such pairs of points are said to be separated by the curve or “linked” with it.
In three-dimensional space, there are certainly no pairs of points which are separated by our Jordan curve, but there are closed polygons which are linked with it in the natural sense that every piece of surface which is bounded by the polygon necessarily has points in common with the curve. Here the portion of the surface spanned by the polygon need not be simply connected but may be chosen entirely arbitrarily.
The Jordan curve theorem may also be generalized in another way for three-dimensional space : in space, there are not only closed curves, but also closed surfaces, and every such surface divides the space into two regions —- exactly as a closed curve did in the plane.
Supported by analogy, the reader can probably imagine what the relationships are in four-dimensional space: for every closed curve, there exists a closed surface linked with it; for every closed three-dimensional manifold a pair of points linked with it. These are special cases of the Alexander duality theorem.
((PS: In one dimension, a manifold may be a straight line, in two dimensions a plane, or the surface of a cube, a balloon, or a doughnut. The defining feature of a manifold is that, from the vantage point of any spot on such an object, the immediate vicinity looks like perfectly regular and normal Euclidean space. Think of yourself shrunk to the size of a pinpoint, sitting on the surface of a doughnut. Look around you, and it seems that you’re sitting on a flat disk. Go down one dimension and sit on a curve, and the stretch nearby looks like a straight line. Should you be perched on a three-dimensional manifold, however esoteric, your immediate neighborhood would look like the interior of a ball. In other words, how the object appears from afar may be quite different from the ,way it appears to your nearsighted eye.
By 1950, topologists were having a field day with manifolds, redefining every object in sight topologically. The diversity and sheer number of manifolds is such that today, although all two-dimensional objects have been defined topologically, not all three- and four-dimensional objects-of which there is literally an infinite assortment- have been so precisely described. Manifolds turn up in a wide variety of physical problems, including some in cosmology, where they are often very hard to cope with. The notoriously difficult three-body problem proposed by King Oskar II of Sweden and Norway in 1885 for a mathematical competition in which Poincar6 took part, which entails predicting the orbits of any three heavenly bodies–such as the sun, moon, and earth-is one in which manifolds figure largely.
This has been a Clay Millennium Problem and has been resolved by Grigory Perelman.))
4. Perhaps, the above examples leave the reader with the impression that in topology nothing at all but obvious things are proved !! 😦
This impression will fade quickly as we go on. However, be that as it may, even these “obvious” things are to be taken much more seriously : one can easily give examples of propositions which sound as “obvious” as the Jordan curve theorem, but which may be proved false. Who would believe, for example, that in a plane there are three (four, firve….in fact, infinitely many!) simply connected bounded regions which all have the same boundary; or that one can find in three-dimensional space a simple Jordan arc (that is, a topological image of a polygonal line) such that there are circles outside of this arc that cannot possibly be contracted to a point without meeting it ? There are also closed surfaces of genus zero which possess an analogous property. In other words, one can construct a topological image of a sphere and an ordinary circle in its interior in such a way that the circle may not be contracted to a point wholly inside the surface.
5. All of these phenomena were wholly unsuspected at the beginning of the 20th century; the development of set theoretic methods in topology first led to their discovery and; consequently, to a substantial extension of our idea of space. However, let me at once issue the emphatic warning that the most important problems of set theoretic topology are in no way confined to the exhibition of, so to speak, “pathological” geometrical structures; on the contrary, they are concerned with something quite positive. I would formulate the basic problem of set theoretic topology as follows:
To determine which set theoretic structures have a connection with the intuitively given material of elementary polyhedral topology and hence, deserve to be considered as geometrical figures —even if very general ones.
Obviously implicit in the formulation of this question is the problem of a systematic investigation of structures of the required type, particularly with reference to those of their properties which actually enable us to recognize the above mentioned connection and so bring about the geometrization of the most general set theoretic topological concepts.
PS: The purpose of sharing or blogging this is just to revise it for myself and hopefully, some readers will also benefit. In particular, these are my own study notes. Especially, Prof Paul Halmos used to say “I write what I talk to myself. I think by writing”…