1. ## Homeomorphism

How do I prove that two closed intervals,
$[a_1,b_1]=\{x \in \Re : a_1 \le x \le b_1\} \ \& \ [a_2,b_2] \ (a_i are homeomorphic? Prove by writing down the formula for a homeomorphism.

2. I didn't say enything

3. Originally Posted by bigdoggy
How do I prove that two closed intervals,
$[a_1,b_1]=\{x \in \Re : a_1 \le x \le b_1\} \ \& \ [a_2,b_2] \ (a_i are homeomorphic? Prove by writing down the formula for a homeomorphism.
Try $f(x) = \left[ {\frac{{b_2 - a_2 }}{{b_1 - a_1 }}} \right]\left( {x - a_1 } \right) + a_2$

4. Originally Posted by Plato
Try $f(x) = \left[ {\frac{{b_2 - a_2 }}{{b_1 - a_1 }}} \right]\left( {x - a_1 } \right) + a_2$
How do you arrive at that!?

5. To prove the homeomorphism you need to find function f:A->B with following properties Homomorphism - Wikipedia, the free encyclopedia.

In your case you have 2 intervals [a1,b1] & [a2,b2]. You can make such function by stretching one interval and then move it to the beginning of the second one.

-------[a1............b1]----------
------------[a2...........b2]------

6. Originally Posted by bigdoggy
How do you arrive at that!?
Well I certainly hope that you recognize that as the equation of a line.
Linear functions are continuous bijections.
This particular line is determined by the pairs $(a_1,a_2)~\&~(b_1,b_2)$
It maps $[a_1,b_1]\mapsto[a_2,b_2]$.