Torus T^2 homeomorphic to S^1 x S^1

[Math Amateur]

I am reading Martin Crossley's book, Essential Topology.

Example 5.43 on page 74 reads as follows:

I am really struggling to get a good sense of why/how/wherefore Crossley came up with the maps f and g in EXAMPLE 5.43. How did he arrive at these maps?

Why/how does f map S^1 \times S^1 onto T^2 and how does one check/prove that this is in fact a valid mapping between these topological spaces.

Can anyone help in making the origins of these maps clear or perhaps just indicate the logic behind their design and construction? I am completely lacking a sense or intuition for this example at the moment ... ...

Definitions for T^2 and S^1 are as follows:

My ideas on how Crossley came up with f and g are totally bankrupt ... but to validate f (that is to check that it actually maps a point of S^1 \times S^1 onto T^2 - leaving out for the moment the concerns of showing that f is a continuous bijection ... ... I suppose one would take account of the fact that (x,y) and (x',y') are points of S^1 and so we have:

x^2 + y^2 = 1 ... ... ... ... (1)

and

x'^2 + y'^2 = 1 ... ... ... ... (2)

Then, keeping this in mind check that

((x' +2)x, (x' +2)y, y' is actually a point on the equation for T^2, namely:

x^2 + y^2 + z^2 - 4 \sqrt{x^2 + y^2} = -3 ... ... ... (3)

So in (3) we must:

- replace x by (x' +2)x
- replace y by (x' +2)y
- replace z by y'

and then simplify and if necessary use (1) (2) to finally get -3.

Is that correct? Or am I just totally confused ?

Can someone please help?

Peter

 

[Mandelbroth]

Look at the terms. What is $(x'+2)x$? What is important about the number 2 here? (Hint: Look at your definition of the torus.)

I've always defined the torus as $S^1\times S^1$. It's kind of "obvious" if you think about it. Just look at this picture: