Guide Elementary topology.A first course

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  1. Topology by James Munkres
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Topology by James Munkres

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Topology: A First Course

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reference request - Textbooks for learning Algebraic Topology - Mathematics Stack Exchange

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Seller Inventory SKU Book Condition is in New International Edition. Same ISBN as per description. Books in English. Quick Order Processing. Perfect Condition. Item added to your basket View basket. A set equipped with a topology is called a topological space. The remainder of this article deals with the branch of mathematics known as topology. Topology has sometimes been called rubber-sheet geometry, because it does not distinguish between a circle and a square a circle made out of a rubber band can be stretched into a square but does distinguish between a circle and a figure eight you cannot stretch a figure eight into a circle without tearing.

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The spaces studied in topology are called topological spaces. They vary from familiar manifolds to some very exotic constructions. It has had a major impact on the fields of differential geometry , algebraic geometry , dynamical systems and partial differential equations in the large, and several complex variables.

What is a Manifold? Lesson 1: Point Set Topology and Topological Spaces

Geometry in the sense of Michael Atiyah and his school now includes all of this. Internally to the subject, point-set topology or general topology is the study of topological spaces without further restrictions; other areas deal with topological spaces that look more like manifolds. These include algebraic topology which grew out of combinatorial topology , geometric topology , low-dimensional topology dealing for example with knot theory , and differential topology. This article is a general overview of topology. For more precise mathematical definitions, see topological spaces or one of the more specialized articles listed below.

The topology glossary contains definitions of terms used throughout topology. The root of topology was in the study of geometry in ancient cultures. Leibniz was the first to employ the term analysus situs, later employed in the 19th century to refer to what is now known as topology. Georg Cantor , the inventor of set theory , had begun to study the theory of point sets in Euclidean space , in the later part of the 19th century, as part of his study of Fourier series.

In , Felix Hausdorff , generalizing the notion of metric space, coined the term "topological space" and gave the definition for what is now called Hausdorff space. Finally, a further slight generalization in , by Kazimierz Kuratowski , gives the present-day concept of topological space. However, Listing had already used the word for ten years in correspondence.

The separate status of the topologist , a specialist in topology, was used in in the magazine Spectator. A toroid in three dimensions; A coffee cup and a donut are both topologically indistinguishable from this toroid. Topological spaces show up naturally in mathematical analysis , abstract algebra and geometry. This has made topology one of the great unifying ideas of mathematics. General topology , or point-set topology , defines and studies some useful properties of spaces and maps, such as connectedness , compactness and continuity.

Algebraic topology is a powerful tool to study topological spaces, and the maps between them. It associates "discrete", more computable invariants to maps and spaces, often in a functorial way. Ideas from algebraic topology have had strong influence on algebra and algebraic geometry. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the "way they are connected together". This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks.