Lindelöf models of the reals: Solution to a problem of Sikorski

We show that it is consistent with ZFC that there is a model M of ZF + DC such that the integers of M are ω ₁-like, the reals of M have cardinality ω ₂, and the unit interval [0, 1] ^M is Lindelöf (i.e. every open cover has a countable subcover). This answers an old question of Sikorski.