Topology

Etymology

The word topology comes from the contraction of the Greek names topos and logos which means place respectively, and study . Literally, topology is the study of the place. It is thus interested to define what is a place (also called space ) and which can be the properties about it.

The topology more precisely is interested in the topological spaces and in the applications which bind them, known as continuous .

It makes it possible to classify these spaces, in particular the nodes , inter alia by their Dimension (which can be as well null as infinite).

It is also interested in their deformations.

In analyzes, thanks to information which it provides on space considered, it makes it possible to obtain a certain number of results (existence and/or unicity of solutions of differential equations , in particular).

The metric spaces as well as the normalized vector spaces are examples of topological spaces.

Intuitive idea

Generally, topology is presented in the form of “Geometrical a rubber sheet” That refers to the Euclidean Géométrie, where two objects are equivalent if one can transform one into the other using Isométrie S (Rotation S, Translation S, reflections, etc….) i.e., of the transformations which preserve the value of the Angle S, of the Length S, the surfaces, of the Volume S and others. In topology, two objects are equivalent in a direction much broader. They must have the same number of pieces, holes, intersections etc…. In topology, it is allowed to double, stretch, twist etc….objects but always without breaking them, neither to separate what is plain, nor to stick what is separate. For example, a triangle is topologically the same thing as a circle, i.e. one can transform one into the other without breaking and without sticking. But a circle is not the same thing that a segment (one must break the circle to obtain the segment). This is why that is called the “Geometry of the rubber sheet” because it is as if one studied the geometry with a rubber sheet which one could contract, stretch, etc

A usual joke between topologists (mathematicians who work on topology) tells that a topologist is a person who cannot distinguish a cup from a fritter.

But this vision, although intuitive and clever, partial and is skewed. On a side, one can think that Topology treats only geometrical objects and concepts (whereas on the contrary, it is the geometry which treats a certain type of topological objects). On another side, in much of case, it is impossible to give the indication of an interpretation of a topological problem, or certain concepts. To try to visualize the concepts is a frequent error at the beginners, who advances them very slowly when they cannot find an example graphic. It is frequent to intend the students to say that they do not include/understand Topology and that they do not like this branch. Generally, one owes this aversion with the fact that the problem cannot be visualized by a drawing. Finally, Topology nourishes also concepts from which the inspiration comes from the mathematical Analysis. One can say that almost the totality of the concepts and ideas of this branch are topological concepts and ideas .

History

The origin of topology is the study of the Géométrie in the ancient cultures. The work of Leonhard Euler going back to 1736 on the Problème of the seven bridges of Königsberg is regarded as one of the first results of geometry which does not depend on any measurement, i.e. one of the first topological results.

Henri Poincaré published Analysis Situs in 1895, introducing the concepts of Homotopie and Homologie.

Maurice Fréchet, unifying work on spaces of functions of Cantor, Volterra, Arzelà, Hadamard, Ascoli and others, introduced the metric concept of Space in 1906.

In 1914, Felix Hausdorff, by generalizing the concept of space metric, invented the term of “ topological Espace ” and defines what is called today the separate Espace or spaces of Hausdorff.

Finally, another light generalization in 1922, by Kuratowski, gave the current concept of topological space.

The term “topology”, was introduced in German in 1847 by Johann Benedict Listing into “ Vorstudien zur Topologie ”.

Appendices

Glossary

Heiner Zieschang
Space topological
metric Space
algebraic Topology
weak Topology
topological Topology network
Mirage

External bond

''' knowledge-without-frontière.com ''' (the topologicon is a popularization of topology in band-drawn written by Jean-Pierre Petit)

Simple: Topology

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