I have been working on a problem and I need to answer the following question:

There is a family $ {F_ alpha: alpha in omega_1 } $ of subsets of the interval $

0.1

$[$[$ such that

(a) $ F_ alpha = {x_1 ^ alpha, x_2 ^ alpha } $ with $ x_1 ^ alpha <x_2 ^ alpha $;

(b) yes $ alpha, beta in omega_1 $ with $ alpha ne beta $, so

$ x_1 ^ alpha <x_1 ^ beta <x_2 ^ alpha <x_2 ^ beta $ or $ x_1 ^ beta <x_1 ^ alpha <x_2 ^ beta <x_2 ^ alpha $?

I have tried to build a family without success. Does anyone see the answer?