A family f₁,..., fₙ of operators on a complete metric space X is called contractive if there exists a positive λ < 1 such that for any x,y in X we have for some i. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. We show that Austin’s conjecture is true for three operators, provided that λ is sufficiently small.
Among other results we prove that the topological statement “Any compact covering mapping between two Π⁰₃ spaces is inductively perfect” is equivalent to the set-theoretical statement ""; and that the statement “Any compact covering mapping between two coanalytic spaces is inductively perfect” is equivalent to “Analytic Determinacy”. We also prove that these statements are connected to some regularity properties of coanalytic cofinal sets in (X), the hyperspace of all compact subsets of a Borel...
If is a space that can be mapped onto a metric space by a one-to-one mapping, then is said to have a weaker metric topology. In this paper, we give characterizations of sequence-covering compact images and sequentially-quotient compact images of spaces with a weaker metric topology. The main results are that (1) is a sequence-covering compact image of a space with a weaker metric topology if and only if has a sequence of point-finite -covers such that for each . (2) is a sequentially-quotient...