# Set theory – Subsets of the interval unit.

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 ;

(b) yes $$alpha, beta in omega_1$$ with $$alpha ne beta$$, so
$$x_1 ^ alpha or $$x_1 ^ beta ?

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