How is a torus described in topology?

[waht]

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?

 

[matt grime]

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.