Proper shape invariants: Tameness and movability (Q1912323)

The author continues the study of invariants in proper shape theory of topological spaces introduced in his article [Acta Sci. Math. 59, No. 3-4, 679-711 (1994; Zbl 0836.54012)]. Here he studies geometric properties related to some forms of proper tameness and proper movability and proves several results on preservation of these properties under proper maps. The origin of the notion of tameness can be traced back to L. Siebenmann's tame at infinity spaces. The approach adopted in this paper consists in requiring that small enough proper multivalued functions from a given space into members of a class of spaces factor through some member of another class also through sufficiently small proper multivalued functions. The notions studied here are related to other forms of proper tameness and movability previously considered by the author and G. M. Gianella. The author shows in the present paper some remarkable duality properties between the two kinds of notions.