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March 19th, 2014, 06:25 PM  #1 
Newbie Joined: Nov 2011 Posts: 9 Thanks: 0  Torus T^2 homeomorphic to S^1 x S^1
[attachment=1:1wfzpmh8]Equation of Torus  Crossley  page 25.png[/attachment:1wfzpmh8]I am reading Martin Crossley's book, Essential Topology. Example 5.43 on page 74 reads as follows: [attachment=2:1wfzpmh8]Crossley  EXAMPLE 5.43  page 74.png[/attachment:1wfzpmh8] 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 onto 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 and are as follows: [attachment=1:1wfzpmh8]Equation of Torus  Crossley  page 25.png[/attachment:1wfzpmh8] [attachment=0:1wfzpmh8]S^1  Crossley  page 32.png[/attachment:1wfzpmh8] 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 onto  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 and so we have: ... ... ... ... (1) and ... ... ... ... (2) Then, keeping this in mind check that is actually a point on the equation for , namely: ... ... ... (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 

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