Countable Products of Čech-scattered supercomplete spaces

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

“…The proof is very similar to the one given in [20] for the case of countably manyČech-scattered paracompacta. Therefore, we will only give the main steps of the proof and refer the reader to [20] for details. Let G be an open cover of ΠX i .…”

“…As in [19], [20], we may extend the proof of 6.1 to the σ-partition-complete case. (Recall that a space is called σ-partition-complete if it is a countable union of partition-complete, closed subspaces.)…”

“…(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.…”

Screenableness in countable products

“…As an excellent result, Friedler et al [5], Hohti and Pelant [7] showed that if {X n : n ∈ ω} is a countable collection of C-scattered paracompact spaces, then the product ∏ n∈ω X n is paracompact. As a generalization of C-scattered spaces,Čech-scattered spaces introduced by Hohti and Ziqiu [8] play the same fundamental role in the study of paracompactness of countable products. In 2005, Aoki and Tanaka [1] extended the Hohti and Ziqiu's results by proving that if Y is a perfect paracompact space, and {X n : n ∈ ω} is a countable collection ofČech-scattered paracompact spaces, then the product Y × ∏ n∈ω X n is paracompact.…”

“…Remark 3.6. If {X n : n ∈ ω} is a countable collection ofČech-scattered screenable spaces, we can assume that X n =X for each n ∈ ω, and X is topped with Top(X)={a} for some a ∈ X, see [1,8]. Therefore, by Theorem 3.4, the product ∏ n∈ω X n is screenable.…”