Reference: Invitation to Combinatorial Topology by Maurice Frechet and Ky Fan, Dover Publications, Inc.
Foreword:
This book, may perchance, fall into the hands of a reader curious about mathematics, but who, before beginning to read our work, wishes to have an idea, expressed in nontechnical language and in a few lines, as to “what is topology (also called analysis situs). Henri Poincare, the founder of topology has had already given satisfactory answer to the reader in an intuitive and particularly striking way:
“Geometers usually distinguish two kinds of geometry, the first of which they qualify as metric and the second as projective. Metric geometry is based on the notion of distance, two figures are there regarded as equivalent when they are ‘congruent’ in the sense that mathematicians give to this word. (like in our Indian maths text books, SSS Test, SAS Test, RHS test…). Projective geometry is based on the notion of a straight line; in order for two figures there to be considered equivalent, it is not necessary that they be congruent; it suffices that one can pass from one to other by projective transformation, that is, that one can be the perspective of the other. This second body of study has often been called qualitative geometry, and in fact, it is if one opposes it to the first; it is clear that measure and quantity ( that is, length, areas, volumes, measures of angles, length of arcs…) play a less important role. This is not entirely so, however, The fact that a line is straight is not purely qualitative; one cannot assure himself that a line is straight without making measurements, or without sliding on this line an instrument called a straightedge, which is a kind of instrument of measure.
“But it is a third geometry from which quantity is completely excluded and which is purely qualitative; this is “analysis situs.” In this figure, two figures are equivalent whenever one can pass from one to the other by a continuous deformation, whatever else the law of this deformation may be, it must be continuous. Thus, a circle is equivalent to an ellipse or even to an arbitrary closed curve, but it is not equivalent to a straight line segment since this segment is not closed. A sphere is equivalent to any convex surface (NB here this word convex has same meaning as in a convex polygon); it is not equivalent to a torus since there is a hole in a torus (or say a donut ! 🙂 and in a sphere there is no hole. Imagine an arbitrary design and a copy of this same design executed by an unskilled draftsman; the properties are altered, the straight line drawn by an inexperienced hand have sufferend unfortunate deviations and contain awkward bends. From the point of view metric geometry, and even of projective geometry, the two figures are not equivalent; on the contary from the point of view of analysis situs, they are equivalent.
Analysis situs is a very important science for the geometer; it leads to a sequence of theorems as well as those of Euclid; and it is not this set of propositions that Riemann has built one of the most remarkable and most abstract theories of pure analysis. I will cite two of these theorems in order to clarify their nature: (1) Two plane closed curves cut each other in an even number of points (2) if a polyhedron is convex, that is, if one cannot draw a closed curve on its surface without dividing it into two, the number of edges is equal to that of the vertices plus that of the faces diminished by two., and this remains true when the faces and edges of the polyhedron are curved.
And here is what makes this analysis situs interesting to us; it is that geometric intution really intervenes there. When, in a theorem of metric geometry, one appeals to this intuition, it is because it is impossible to study the metric properties of a figure as abstractions of its qualitative properties, that is, of those which are the proper business of analysis situs. It has often been said that geometry is the art of reasoning correctly from badly drawn figures. This is not a capricious statement; it is a truth that merits reflection. But what is a badly drawn figure ? It is what might be executed by the unskilled draftsman spoken of earlier; he alters the properties more or less grossly; his straight lines have disquieting zigzags; his circles show awkward bumps. But this does not matter; this will by no means bother the geometer; this will not prevent him from reasoning correctly.
“But the inexperienced artist must not represent a closed curve as an open curve, three lines which intersect in a common point by three lines which have no common point, or a pierced surface by a surface without a hole. Because then one would no longer be able to avail himself of his figure and reasoning would become impossible. Intuition will not be impeded by the flaws of drawing which concern only metric or projective geometry; it becomes impossible as soon as these flaws relate to analysis situs.
“This very simple observation shows us the real role of geometric intuition; it is to assist this intuition that the geometer needs to draw figures, or at the very least to represent them mentally. Now, if he ignores the metric or projective properties of these figures, if he adheres only to their purely qualitative properties, it will be then and only then that geometric intuition really intervenes. Not that I wish to say that metric geometry rests on pure logic and that true intuition never intervenes there, but these are intuitions of a different kind, analogous to those which play the essential role in arithmetic and in algebra.”
Cheers,
Nalin Pithwa