# Thread: How to find homeomorphism between sets

1. ## How to find homeomorphism between sets

Find homeomorphism between sets
1 - $\displaystyle \left[ {a,\infty } \right[\& \left[ { - 1,1} \right[$
2- $\displaystyle \left] { - \infty ,1} \right]\& \left] { - 1,1} \right]\,$

is the solution

$\displaystyle \begin{array}{l} f_1 \left( x \right) = \frac{{x - 2a}}{x} \\ f_2 \left( x \right) = \frac{{2 - x}}{x} \\ \end{array}$

2. ## Re: How to find homeomorphism between sets

Originally Posted by rqeeb
Find homeomorphism between sets
1 - $\displaystyle \left[ {a,\infty } \right[\& \left[ { - 1,1} \right[$
2- $\displaystyle \left] { - \infty ,1} \right]\& \left] { - 1,1} \right]\,$

is the solution

$\displaystyle \begin{array}{l} f_1 \left( x \right) = \frac{{x - 2a}}{x} \\ f_2 \left( x \right) = \frac{{2 - x}}{x} \\ \end{array}$
These are good attempts, but there is a snag. The solution for 1) is correct provided that a>0. But it does not work if a<0 because in that case x can take the value 0, and the function $\displaystyle f_1(x)$ has a vertical asymptote there.

The solution to 2) fails for the same reason – it goes wrong at x=0. But you can easily put that right in this case by flipping the function over and defining $\displaystyle f_2(x) = \tfrac x{2-x}.$ Since x does not take the value 2, the denominator is not going to vanish.

TeX tip: the TeX engine used in this forum does not like line breaks. If you write your TeX input all on one line, as

[TEX]\begin{array}{l} f_1 \left( x \right) = \frac{{x - 2a}}{x} \\ f_2 \left( x \right) = \frac{{2 - x}}{x} \\ \end{array}[/TEX] ,

it will be rendered without the tiresome $\displaystyle$s.

3. ## Re: How to find homeomorphism between sets

yes .. u r right