Reference: Elementary Concepts of Topology by Paul Alexandroff, Dover Publications. (Available Amazon India):

Introduction:

- 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. (Note one can also view topology as an abstract study of the concept of the limit point because so many fundamental mathematical ideas like continuity and connectedness depend on it. We can also say that algebra and topology are complementary mathematical operations. Further that, whereas in Euclidean geometry, we consider rigid motions of geometric objects and that lengths, areas, volumes, angles are to be invariant, that congruence and similarity are geometric properties; in topology, which can be considered rubber sheet geometry, a geometric object, example a polyhedron can be stretched, twisted, bent, pulled; cut and coalesced exactly same way as before; but no holes can be destroyed or made; topological properties of a geometric object are those which are invariant under these transformations. For example, that a point lies in the interior of a circle is a topological property). Professor Alexandroff had said further in the above reference:

A. It is impossible to map an n-dimensional cube onto a proper subset of itself by a continuous deformation in which the boundary remains point-wise fixed. (Comment: to “verify” or “visualize” this, just try to sketch a one-to-one correspondence of a Rubik’s cube onto any face of it. The cube is three dimensional whereas the face is 2 dimensional).

That this seemingly obvious theorem is in reality a very deep one can be seen from the fact that from it follows the invariance of dimension (that is, the theorem that it is impossible to map two coordinate spaces of different dimensions onto one another in a one-to-one and bicontinuous fashion.)

The invariance of dimension may also be derived from the following theorem which is among the most beautiful and most intuitive of topological results:

B. The Tiling Theorem: If one “covers” an n-dimensional cube with finitely many sufficiently small (but otherwise entirely arbitrary) closed sets, then there are necesssarily points which belong to at least n+1 of these sets. (On the other hand, there exists arbitrarily fine coverings for which this number n+1 is not exceeded). (Comment: Recall definition of “covering”, “open covering” and a “compact set”). (Digressing, there is the well-known example of fixed point theorems or of the well-known topological properties of closed surfaces such as are described in Hilbert and Cohn-Vossen’s Geometry and the Imagination, Chelsea Publications) (NB: here, sufficiently small will mean with a sufficiently small diameter).

For n=2, the theorem asserts that if a country is divided into sufficiently small provinces,there necessarily exist points at which at least three provinces come together. Here these provinces may have entirely arbitrary shapes, in particular, they need not even be connected (that is, in the language of analysis or topology, they are “separated” sets); each one may consist of several pieces.

Recent(at that time circa. 1905) topological investigations have shown that the whole nature of the concept of dimensions is hidden in this covering or tiling property, and thus the tiling theorem has contributed in a significant way to the deepening of our understanding of space.

C. As the third example of an important and yet obvious-sounding theorem, we may choose 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.

II. The question which naturally arises is: What can one say about a closed Jordan curve 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 can 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, you can probably imagine what the relationships are in four-dimensional space: for every closed curve 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 which we will discuss later.

III. Perhaps the above examples leave you with the impression that in topology nothing at all but obvious things are proved; this impression will fade rather 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 (in fact, four, five, six…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 it 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.

IV. All of these phenomena were wholly unsuspected at the beginning of the century (20th century): the development of set theoretic methods in topology first led to their discovery and consequently,to a substantial extention 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. Prof Alexandroff had formulated 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.

The program of investigation for set-theoretic topology thus formulated is to be considered — at least in basic outline — as completely capable of being carried out; it has turned out that the most important parts of set theoretic topology are amenable to the methods which have been developed in polyhedral topology. Thus, it is justified in what follows we devote ourselves primarily to the topology of polyhedra. (Ref: next blog article).

Cheers,

Nalin Pithwa