Hausdorff compactness on a space of ω-limit sets

In [Trans. Amer. Math. Soc. 348 (4) (1996)], Blokh et al., studied the space of the ω-limit sets W ( f ) , produced by a continuous map on I = [ 0 , 1 ] , and established that endowed with the Hausdorff metric topology on I, this space is compact. For general continuous maps on I 2 , we show that this space is not compact and for maps whose W ( F ) is included in a fiber of I 2 , we present examples of both types: holding and not holding the property of being compact.