"Topology, or analysis situs, is a modern branch of geometry which […] does not bring in the notions of size or measure, but only that of continuity. It concerns itself, then, only with qualitative properties of figures. More precisely, one can define the aim of topology as follows. A property of a set is said to be topological if it can be expressed by means of the concept of continuity. A topological property of a set is called a topological invariant if it is preserved under all homeomorphisms. Topology is the study of topological properties and, especially, topological invariants of figures." (Maurice Frechet & Ky Fan, "Initiation to Combinatorial Topology", 1967)
"An initial study of tensor analysis can. almost ignore the topological aspects since the topological assumptions are either very natural" (continuity, the Hausdorff property) or highly technical" (separability, paracompactness). However, a deeper analysis of many of the existence problems encountered in tensor analysis requires assumption of some of the more difficult-to-use topological properties, such as compactness and paracompactness. " (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)
"From its beginning critical point theory has been concerned with mutual relations between topology and geometric analysis, including differential geometry. Although it may have seemed to many to have been directed in its initial years toward applications of topology to analysis, one now sees that the road from topology to geometric analysis is a two-way street. Today the methods of critical point theory enter into the foundations of almost all studies of analysis or geometry 'in the large'." (Marston Morse & Stewart S Cairns, "Critical Point Theory in Global Analysis and Differential Topology: An Introduction", 1969)
"Mathematicians are finding that the study of global analysis or differential topology requires a knowledge not only of the separate techniques of analysis, differential geometry, topology, and algebra, but also a deeper understanding of how these fields can join forces." (Marston Morse & Stewart S Cairns, "Critical Point Theory in Global Analysis and Differential Topology: An Introduction", 1969)
"Though combinatorics has been successfully applied to many branches of mathematics these can not be compared neither in importance nor in depth to the applications of analysis in number theory or algebra to topology, but I hope that time and the ingenuity of the younger generation will change this." (Paul Erdős, "On the application of combinatorial analysis", Proceedings of the International Congress of Mathematicians Nice, 1970)
"Topology, which used to be called geometry of situation or analysis situs ('topos' means position, situation in Greek), considers that all pots with two handles are of the same form because, if both are infinitely flexible and compressible, they can be molded into any other continuously, without tearing any new opening or closing up any old one. It also teaches that all single island coastlines are of the same form, because they are topologically identical to a circle." (Benoît B Mandelbrot, "The Fractal Geometry of Nature" 3rd Ed., 1983)
"Modem geometry and topology take a special place in mathematics because many of the objects they deal with are treated using visual methods. […] Each mathematician has his own system of concepts of the intrinsic geometry of his" (specific) mathematical world and visual images which he associated with some or other abstract concepts of mathematics" (including algebra, number theory, analysis, etc.). It is noteworthy that sometimes one and the same abstraction brings about the same visual picture in different mathematicians, but these pictures born by imagination are in most cases very difficult to represent graphically, so to say, to draw." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)
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