On perfectness of Zaitov metrization of the hyperspace functor

Recently, A. Zaitov suggested a new metric on the space of all nonempty compact subsets of a given metric space. In the present paper we show that this metric turns the hyperspace functor exp into a perfect metrizable functor. Further, we establish that the hyperspace functor has many remarkable properties with respect to this metrization. In particular, this functor preserves isometric embeddings.