On complexity of metric spaces

Countable Products of Čech-scattered supercomplete spaces

“…To describe the trees we shall deal with, we start from theČ -decomposition tree TČ (X) of X. As in [18] and [19], let End(T ) denote the set of all maximal elements of any tree. Recall that for each point P of TČ (X) the immediate successors of P , P ∈ End(TČ (X)), are the Čech-complete closed subset topČ (P ) of P and the closed sets Q ⊂ P − topČ (P ) such that 1) RankČ (Q) < RankČ (P ) and 2) the interiors int(Q) in P are non-empty.…”

“…If K is a closed-hereditary class, then (in this paper all spaces are assumed to be at least Tychonoff spaces) we can define the corresponding K -decomposition tree T K (X) of X as follows. (For terminology concerning trees, the reader is kindly referred to [18] or [19].) The elements of T K (X) are closed subsets of X defined by using the derivative sets D…”

The locally fine coreflection and normal covers in the products of partition-complete spaces

“…Well-founded trees in the context of locally fine refinements were first used in [31] and in an early version of [30]. They were consequently applied in several papers: [18], [19], [20], and in [17] in the context of well-founded cubical triangulations refining open covers of infinite-dimensional cubes.…”

“…For supercomplete spaces [22], and in particular for complete metric spaces, this procedure reaches all open covers of the space. The complexity of such refinements was studied in [12] and [18]. Related methods were used by the second and third authors to obtain extension theorems for continuous functions defined on subsets of products of metrizable spaces [21].…”