Topological properties of hyperspaces

In the work, it is given the normal functor F acting in the category of compacts and their continuous mappings. This functor does not preserve the Souslin number (or hereditary cellularity), the hereditary density, the hereditary π-weight and the hereditary Shanin number, the hereditary caliber, the hereditary precaliber, the hereditary preshanin number, the hereditary weak density, the hereditary Lindelöf number, and the hereditary extent of a compact. The example of the normal functor and the compact of Aleksandorv’s two arrows are given. We study the action of functors expn, expω, expc and exp on finally compact, hereditarily disconnected, strongly zero-dimensional, extremely disconnected, paracompact spaces.