On compact fibered spaces

A space X is called fibered if there exists a countable family γ of sets closed in X such that γ(x)=⋂{F: x∈F∈γ} is metrizable for each x∈X. In the paper we answer two problems of Tkachuk raised in [Topology Proc. 19 (1994) 321–334] about compact fibered spaces.