Monotone versions of countable paracompactness

Abstract

One possible natural monotone version of countable paracompactness, MCP, turns out to have some interesting properties. We investigate various other possible monotonizations of countable paracompactness and how they are related.

Monotone versions of δ-normality

“…(3) There is an operator ψ as in (2) such that, in addition, ψ(x, U ) ⊆ U . (4) There is an operator ψ as in (2) such that, in addition, ψ(x, U ) ⊆ U . Definition 4 Let X be a space and C be a collection of pairs of disjoint closed sets.…”

“…In [4], the current authors consider various other possible monotone versions of countable paracompactness and the notion of mδn (monotone δ-normality) arises naturally in this study. It turns out that MCP and mδn are distinct properties and that, if X × [0, 1] is mδn, then X (and hence X × [0, 1]) is MCP.…”

Function characterizations of some spaces in which compacta are Gδ

“…It is known that many classes of spaces such as stratifiable spaces [5,6], k-semi-stratifiable space [8,15], countably paracompact spaces [9,16], monotonically countably paracompact spaces [3] can be characterized with real-valued functions that satisfy certain conditions. In [13], to give characterizations of some generalized metric spaces, the following conditions were introduced.…”

Monotone partitions and almost partitions

“…For details, see e.g., [2,[4][5][6][7][8]12,14]. A space is zero-dimensional if it has a base consisting of open-and-closed (abbreviated clopen) sets.…”