On supercomplete uniform spaces IV: Countable products

Countable Products of Čech-scattered supercomplete spaces

“…On the other hand, the earlier result of Telgársky [30] on finite products of C-scattered paracompact spaces was extended to the stronger, combinatorial condition of supercompleteness [21] by the first author in [16]. Following this research line ( [14], [16], [17], [26]), the first author and J. Pelant gave in [19] the result establishing the supercompleteness of countable products of C-scattered supercomplete spaces, which implies the above-mentioned results of [1], [6], [5], and [28]. However, Frolík [6] had also proved (more generally) that the countable product of Čech-complete paracompact spaces is paracompact.…”

“…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…”

“…This section is divided into four subsections. We adopt many definitions from [10], [19], in particular the definitions of well-founded tree and the locally finite (Ginsburg-Isbell) coreflection operator λ. We consider in this paper trees as partially ordered sets T with the underlying set denoted by the same symbol.…”

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

“…(Notice the relation with the so-called θ-refinability of (topological) spaces). The metric-fine coreflection was used in both [19] and [20] to extend the results from K-scattered spaces to σ − K-scattered, where K was the class of compact (resp.Čech-scattered paracompact) spaces. The metric-fine coreflection m can be directly extended to pre-uniformities of topological spaces.…”