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In Munkres book "Topology" (Second Edition), Munkres proves that a function F is a homeomorphism ...
I need help in determining how to find the inverse of $$F$$ ... so that I feel I have a full understanding of all aspects of the example ...
Example 5 reads as follows:View attachment 4193Wishing to understand all aspects of the problem I tried to see how given
$$ F(x) = \frac{x}{1 - x^2} $$
one could determine the inverse of $$F$$ (and then come up with $$G$$, as Munkres did ... ... somehow ??) ... ...I think I proceed by putting
$$y = \frac{x}{1 - x^2}$$
and solving for $$x$$ ... ... BUT how exactly do I proceed (I got nowhere with this problem!)Can someone please help?NOTE... I realize that being able to determine the inverse of F and show it is G is not strictly necessary in showing that F is a homeomorphism ... BUT ... I feel very dissatisfied that I cannot see exactly how this works ... so, again, I hope someone can help ...
Peter
I need help in determining how to find the inverse of $$F$$ ... so that I feel I have a full understanding of all aspects of the example ...
Example 5 reads as follows:View attachment 4193Wishing to understand all aspects of the problem I tried to see how given
$$ F(x) = \frac{x}{1 - x^2} $$
one could determine the inverse of $$F$$ (and then come up with $$G$$, as Munkres did ... ... somehow ??) ... ...I think I proceed by putting
$$y = \frac{x}{1 - x^2}$$
and solving for $$x$$ ... ... BUT how exactly do I proceed (I got nowhere with this problem!)Can someone please help?NOTE... I realize that being able to determine the inverse of F and show it is G is not strictly necessary in showing that F is a homeomorphism ... BUT ... I feel very dissatisfied that I cannot see exactly how this works ... so, again, I hope someone can help ...
Peter