Decomposition, Mapping, and Sum Theorems of ω-Paracompact Topological Spaces - Samer Al Ghour (Axioms, 2021)
As a weaker form of <semantics>ω</semantics>-paracompactness, the notion of <semantics>σ</semantics>-<semantics>ω</semantics>-paracompactness is introduced. Furthermore, as a weaker form of <semantics>σ</semantics>-<semantics>ω</semantics>-paracompactness, the notion of feebly <semantics>ω</semantics>-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly <semantics>ω</semantics>-paracompact. Furthermore, it is proven hereinthat countably <semantics>ω</semantics>-paracompact <semantics>σ</semantics>-<semantics>ω</semantics>-paracompact topological spaces are <semantics>ω</semantics>-paracompact. Furthermore, it is proven hereinthat <semantics>σ</semantics>-<semantics>ω</semantics>-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat <semantics>ω</semantics>-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if <semantics>A</semantics> is a locally finite closed covering of a topological space <semantics><mfenced separators="" open="(" close=")">X,τ</mfenced></semantics> with each <semantics>A∈A</semantics> being <semantics>ω</semantics>-paracompact and normal, then <semantics><mfenced separators="" open="(" close=")">X,τ</mfenced></semantics> is <semantics>ω</semantics>-paracompact and normal, and as a corollary, a sum theorem for <semantics>ω</semantics>-paracompact normal topological spaces follows. Moreover, three open questions are raised.