Abstract

We study geometric properties of topological spaces called proper NCℬ-tameness, proper PCℬ-tameness, and proper Nℬ-movability, where ℬ and C denote classes of spaces. They are related to proper MCℬ-tameness and proper Mℬ-movability from [5] and could be regarded as their dual forms. All three are invariants of a recently invented author's proper shape theory and are described by the use of proper multi-valued functions. We explore their basic properties and prove several results on their preservation under proper maps.