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摘要


We study geometric properties of topological spaces called proper N_C^B-tameness, proper P_C^B-tameness, and proper N^B-movability, where B and C denote classes of spaces. They are related to proper M_C^B-tameness and proper M^B-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.

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