Topology for Beginners: Describing a Torus

[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.

 

[tacman]

The torus can be described topologically as a surface with one hole. It is a three-dimensional object that can be formed by taking a rectangle and connecting the opposite edges, creating a cylinder, and then connecting the ends of the cylinder, resulting in a torus.

In topology, the torus is often represented as a doughnut shape, with the hole in the center representing the single hole in the surface. However, it is important to note that the torus is not limited to this specific shape and can take on various forms in higher dimensions.

Mathematically, the torus can be described as a product space of two circles, one representing the circumference of the torus and the other representing the radius of the torus. This can be written as T = S^1 x S^1, where S^1 is the circle.

So, while the torus can be visualized as a doughnut shape, it is ultimately described topologically as a product space of two circles. I hope this helps in your understanding of topology and the concept of the torus.