Topology for Beginners: Describing a Torus

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SUMMARY

The discussion focuses on the topological description of a torus, highlighting two primary models: the "flat" torus and the "complex" torus. The flat torus is defined as the set of all points in R^2 modulo the relation (x,y)~(u,v) if x-u and y-v are integers, utilizing the quotient topology. In contrast, the complex torus is represented as the product of two circles of radius 1, denoted as S^1 x S^1, with the product topology and the natural subspace topology from C (or R^2). Both models are established as homeomorphic, demonstrating their equivalence in topology.

PREREQUISITES
  • Understanding of basic topology concepts, including homeomorphism.
  • Familiarity with quotient topology and its applications.
  • Knowledge of R^2 and its geometric properties.
  • Basic understanding of complex numbers and their representation in R^2.
NEXT STEPS
  • Research the concept of homeomorphism in topology.
  • Explore quotient topology and its implications in different geometric contexts.
  • Learn about the properties of S^1 and its role in defining complex structures.
  • Investigate algebraic geometry and its relationship with topological spaces.
USEFUL FOR

This discussion is beneficial for students and enthusiasts of topology, particularly those interested in geometric modeling and the foundational concepts of algebraic geometry.

waht
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I read how in topology you can bend a ractangle into a cylinder and then the cylinder into a torus. I'm a beginner to topology, so how is the torus described topologically? Is it just a set of all points? or an equation?
 
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How are those different? That is the basis of algebraic geometry.

There are several ways ot model a torus.

There is the "flat" torus, which is the set of all points in R^2 modulo the relation (x,y)~(u,v) iff x-u and y-v are integers, with the quotient topology.

Then there is the "complex" torus which is the product of two circles of radius 1: S^1xS^1 with the product toplogy on it, adn the natural subspace topologyon S^1 thought of as a subset of C (or R^2). This is a subspace of C^2 or R^4 and the topology is also the same as the subspace topology.

These are all homeomorphic.
 

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