Reference: Elementary Concepts of Topology by Paul Alexandroff, Dover Publications Inc., available in Amazon India:
Preface: David Hilbert, Gottingen, June 1932.
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 superficially 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 present text of Paul Alexandroff is to be greeted as a welcome complement to my Anschauliche Geometrie on the side of of topological systematization; may it win new friends for the science of geometry.
Remarks : (NKP): PS: In this “brief summary” sort of text, Prof. Alexandroff deals shows how geometric intuition is formalized by concentrating on complex, cycle and homology. He shows how Betti groups may be the starting point to apply topological concepts to other fields.
Cheers, cheers, cheers,
Nalin Pithwa
MScMathematics 