On Soft Separation Axioms and Their Applications on Decision-Making Problem

Abstract

In this work, we introduce new types of soft separation axioms called <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>p</a:mi><a:mi>t</a:mi></a:math> -soft <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:mi>α</c:mi></c:math> regular and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"><e:mi>p</e:mi><e:mi>t</e:mi></e:math> -soft <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M4"><g:mi>α</g:mi><g:msub><g:mrow><g:mi>T</g:mi></g:mrow><g:mrow><g:mi>i</g:mi></g:mrow></g:msub></g:math> -spaces <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M5"><i:mfenced open="(" close=")" separators="|"><i:mrow><i:mi>i</i:mi><i:mo>=</i:mo><i:mn>0,1,2,3,4</i:mn></i:mrow></i:mfenced></i:math> using partial belong and total nonbelong relations between ordinary points and soft <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" id="M6"><n:mi>α</n:mi></n:math> -open sets. These soft separation axioms enable us to initiate new families of soft spaces and then obtain new interesting properties. We provide several examples to elucidate the relationships between them as well as their relationships with <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" id="M7"><p:mi>e</p:mi></p:math> -soft <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" id="M8"><r:msub><r:mrow><r:mi>T</r:mi></r:mrow><r:mrow><r:mi>i</r:mi></r:mrow></r:msub></r:math> , soft <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M9"><t:mi>α</t:mi><t:msub><t:mrow><t:mi>T</t:mi></t:mrow><t:mrow><t:mi>i</t:mi></t:mrow></t:msub></t:math> , and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M10"><v:mi>t</v:mi><v:mi>t</v:mi></v:math> -soft <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M11"><x:mi>α</x:mi><x:msub><x:mrow><x:mi>T</x:mi></x:mrow><x:mrow><x:mi>i</x:mi></x:mrow></x:msub></x:math> -spaces. Also, we determine the conditions under which they are equivalent and link them with their counterparts on topological spaces. Furthermore, we prove that <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M12"><z:mi>p</z:mi><z:mi>t</z:mi></z:math> -soft <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" id="M13"><bb:mi>α</bb:mi><bb:msub><bb:mrow><bb:mi>T</bb:mi></bb:mrow><bb:mrow><bb:mi>i</bb:mi></bb:mrow></bb:msub></bb:math> -spaces <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" id="M14"><db:mfenced open="(" close=")" separators="|"><db:mrow><db:mi>i</db:mi><db:mo>=</db:mo><db:mn>0,1,2,3,4</db:mn></db:mrow></db:mfenced></db:math> are additive and topological properties and demonstrate that <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" id="M15"><ib:mi>p</ib:mi><ib:mi>t</ib:mi></ib:math> -soft <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" id="M16"><kb:mi>α</kb:mi><kb:msub><kb:mrow><kb:mi>T</kb:mi></kb:mrow><kb:mrow><kb:mi>i</kb:mi></kb:mrow></kb:msub></kb:math> -spaces <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" id="M17"><mb:mfenced open="(" close=")" separators="|"><mb:mrow><mb:mi>i</mb:mi><mb:mo>=</mb:mo><mb:mn>0,1,2</mb:mn></mb:mrow></mb:mfenced></mb:math> are preserved under finite product of soft spaces. Finally, we discuss an application of optimal choices using the idea of <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" id="M18"><rb:mi>p</rb:mi><rb:mi>t</rb:mi></rb:math> -soft <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" id="M19"><tb:msub><tb:mrow><tb:mi>T</tb:mi></tb:mrow><tb:mrow><tb:mi>i</tb:mi></tb:mrow></tb:msub></tb:math> -spaces <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" id="M20"><vb:mfenced open="(" close=")" separators="|"><vb:mrow><vb:mi>i</vb:mi><vb:mo>=</vb:mo><vb:mn>0,1,2</vb:mn></vb:mrow></vb:mfenced></vb:math> on the content of soft weak structure. We provide an algorithm of this application with an example showing how this algorithm is carried out. In fact, this study represents the first investigation of real applications of soft separation axioms.