| Summary: | As a weaker form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompactness, the notion of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompactness is introduced. Furthermore, as a weaker form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompactness, the notion of feebly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompact. Furthermore, it is proven hereinthat countably <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompact <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompact topological spaces are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompact. Furthermore, it is proven hereinthat <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> is a locally finite closed covering of a topological space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>X</mi><mo>,</mo><mi>τ</mi></mfenced></semantics></math></inline-formula> with each <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="script">A</mi></mrow></semantics></math></inline-formula> being <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompact and normal, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>X</mi><mo>,</mo><mi>τ</mi></mfenced></semantics></math></inline-formula> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompact and normal, and as a corollary, a sum theorem for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-paracompact normal topological spaces follows. Moreover, three open questions are raised.
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