I'm working on a problem and I have to answer the following question:

Is there a family $ {F_ alpha: alpha in omega_1 $ subsets of the interval $]0.1[$[$[$[$ such as:

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

b) If $ alpha, beta in omega_1 $ with $ alpha ne beta $then

$ 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 such a family without success. Does anyone see the answer?