"The stable manifolds of the critical points of a nice function can be thought of as the cells of a complex while the unstable manifolds are the dual cells. This structure has the advantage over previous structures that both the cells and the duals are differentiably imbedded in M. We believe that nice functions will replace much of the use of С triangulations and combinatorial methods in differential topology." (Steven Smale, "The generalized Poincare conjecture in higher dimensions", Bull. Amer. Math. Soc. 66, 1960)
"Certainly, the problems of combinatorial manifolds and the relationships between combinatorial and differentiable manifolds are legitimate problems in their own right. An example is the questionof existence and uniqueness of differentiable structures on a combinatorial manifold. However, we don't believe such problems are the goal of differential topology itself. This view seems justified by the fact that today one can substantially develop differential topology most simply without any reference to the combinatorial manifolds." (Steven Smale, "A survey of some recent developments in differential topology", 1961)
"It has turned out that the main theorems in differential topology did not depend on developments in combinatorial topology. In fact, the contrary is the case; the main theorems in differential topology inspired corresponding ones in combinatorial topology, or else have no combinatorial counterpart as yet (but there are also combinatorial theorem whose differentiable analogues are false)." (Steven Smale, "A survey of some recent developments in differential topology", 1961)
"[...] it is clear that differential geometry, analysis and physics prompted the early development of differential topology (it is this that explains our admitted bias toward differential topology, that it lies close to the main stream of mathematics). On the other hand, the combinatorial approach to manifolds was started because it was believed that these means would afford a useful attack on the differentiable case." (Steven Smale, "A survey of some recent developments in differential topology", 1961)
"We consider differential topology to be the study of differentiable manifolds and differentiable maps. Then, naturally, manifolds are considered equivalent if they are diffeomorphic, i.e., there exists a differentiable map from one to the other with a differentiable inverse." (Steven Smale, "A survey of some recent developments in differential topology", 1961)
"In all candor, we must admit that the intuitive meaning of compactness for topological spaces is somewhat elusive. This concept, however, is so vitally important throughout topology […]"
"Topology provides the synergetic means of ascertaining the values of any system of experiences. Topology is the science of fundamental pattern and structural relationships of event constellations." (R Buckminster Fuller, "Operating Manual for Spaceship Earth", 1963)
"Rational mechanics is mathematics, just as geometry is mathematics. […] Mechanics cannot, any more than geometry, exhaust the properties of the physical universe. […] Mechanics presumes geometry and hence is more special; since it attributes to a sphere additional properties beyond its purely geometric ones, the mechanics of spheres is not only more complicated and detailed but also, on the grounds of pure logic, necessarily less widely applicable than geometry. This, again, is no reproach; geometry is not despised because it is less widely applicable than topology. A more complicated theory, such as mechanics, is less likely to apply to any given case; when it does apply, it predicts more than any broader, less specific theory."
"In every subject one looks for the topological and algebraic structures involved, since these structures form a unifying core for the most varied branches of mathematics." (K Weise and H Noack, "Aspects of Topology" 2nd Ed. , 1967)
"A manifold can be given by specifying the coordinate ranges of an atlas, the images in those coordinate ranges of the overlapping parts of the coordinate domains, and the coordinate transformations for each of those overlapping domains. When a manifold is specified in this way, a rather tricky condition on the specifications is needed to give the Hausdorff property, but otherwise the topology can be defined completely by simply requiring the coordinate maps to be homeomorphisms." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)
"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)
"General or point set topology can be thought of as the abstract study of the ideas of nearness and continuity. This is done in the first place by picking out in elementary geometry those properties of nearness that seem to be fundamental and taking them as axioms." (Andrew H Wallace, "Differential Topology: First Steps", 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)
"In mathematics, logic, linguistics, and other abstract disciplines, the systems are not assigned to objects. They are defined by an enumeration of the variables, their admissible values, and their algebraic, topological, grammatical, and other properties which, in the given case, determine the relations between the variables under consideration." (George Klir, "An approach to general systems theory", 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)
"To abstract the qualitative features of a differential equation on M, the concept of a phase portrait become important. Usually the phase portrait means the picture of the solution curves of the differential equation. [...] Then two differential equations on M have the same phase portrait if they are topologically equivalent. A definition of phase portrait is thus a topological equivalence class of differential equations on M. A main goal of the qualitative study of ordinary differential equations is to obtain information on the phase portrait of differential equations." (Steven Smale, "What is global analysis?", American Mathematical Monthly Vol. 76 (1), 1969)
"[…] topology, a science that studies the properties of geometric figures that do not change under continuous transformations.
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