(O-C)-compact spaces and hyperspaces functor

In the work, it is established that the space of all nonempty compact subsets of a Tychonoff space is (O-C) -compact if and only if the give Tychonoff space is (O-C) -compact. Further, for a map f:X→Y the map expβX→Y is (O-C) -compact if and only if the map f is (O-C) -compact.