A Topological Aperitif

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Springer Science & Business Media, Apr 3, 2009 - Mathematics - 152 pages
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Topologyhasbeenreferredtoas“rubber-sheetgeometry”.Thenameisapt,for the subject is concerned with properties of an object that would be preserved, no matter how much it is stretched, squashed, or distorted, so long as it is not in any way torn apart or glued together. One’s ?rst reaction might be that such animprecise-soundingsubjectcouldhardlybepartofseriousmathematics,and wouldbeunlikelytohaveapplicationsbeyondtheamusementofsimpleparlour games. This reaction could hardly be further from the truth. Topology is one of the most important and broad-ranging disciplines of modern mathematics. It is a subject of great precision and of breadth of development. It has vastly many applications, some of great importance, ranging from particle physics to cosmology, and from hydrodynamics to algebra and number theory. It is also a subject of great beauty and depth. To appreciate something of this, it is not necessary to delve into the more obscure aspects of mathematical formalism. For topology is, at least initially, a very visual subject. Some of its concepts apply to spaces of large numbers of dimensions, and therefore do not easily submit to reasoning that depends upon direct pictorial representation. But even in such cases, important insights can be obtained from the visual - rusal of a simple geometrical con?guration. Although much modern topology depends upon ?nely tuned abstract algebraic machinery of great mathematical sophistication, the underlying ideas are often very simple and can be appre- ated by the examination of properties of elementary-looking drawings.
 

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Contents

Homeomorphic Sets
1
Topological Properties
15
Equivalent Subsets
24
Surfaces and Spaces
51
Polyhedra
69
Winding Number
93
Continuity
105
Knots
113
History
121
Solutions
127
Bibliography
149
Index
151
Copyright

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About the author (2009)

DAVID JORDAN is president of the Philadelphia Athletics Historical Society and also lectures frequently on both baseball and the Civil War. He is the author of Occasional Glory: The History of the Philadelphia Phillies; The Athletics of Philadelphia: Connie Mack's White Elephants, 1901-1954; and A Tiger in His Time: Hal Newhouser and the Burden of Wartime Baseball.

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