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topology





topology
Topology explores the properties of geometric shapes that have been deformed but not torn. If A1 and B1 are folded so that the arrows point in the same direction, a collar A2 and a twisted band B2 result. A2 has an inner and outer surface and two edges, but B2 has only one of each – precisely because of its twist. B2 is called a Möbius band and, unlike A2, a path beginning at S and traced round the band will, after one revolution, end on the other "side" of the band at T. A Möbius band stretched as in C1 is distorted further into one half of a Klein bottle (C2), which again has only one edge and one surface. Compare with a normal bottle (C3).
The study of those properties of mathematical objects that remain unaffected by smooth deformations, such as stretching and squeezing, but that don't involve tearing.

The word comes from the Greek topos for "place," and was introduced into English by Solomon Lefschetz in the late 1920s.

A topologist has been described as someone who doesn't know the difference between a doughnut and a coffee cup. Substitute "care about" for "know" and this becomes more accurate.

Imagine a doughnut-shape, or torus, made of soft clay. A potter can easily shape this into a cup with a handle without removing or creating any new holes. Both shapes, in topology, are said to be genus 1 – objects with a single hole. A sphere, by contrast, is genus 0 (no holes), while an eyeglass frame, with the lenses removed, is genus 2.

For more on topologically intriguing structures, see Möbius band and Klein bottle.


Related entry

   • algebraic topology


Related categories

   • TOPOLOGY
   • SPACE AND TIME