Hausdorff compactness on a space of ω-limit sets

AbstractIn [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 I2, we show that this space is not compact and for maps whose W(F) is included in a fiber of I2, we present examples of both types: holding and not holding the property of being compact