A Study About (γ, γʹ)^α–Regular Spaces, Normal Spaces with (γ, γʹ)–Open Sets and α_{(γ, γ}ʹ_)–Open Sets

Abstract

In the present article, the idea of $\left( {{\gamma }, {\gamma }^{\prime} } \right)^{\alpha }$ -regular spaces and $\left( {{\gamma }, {\gamma }^{\prime} } \right)^{\alpha }$-normal spaces has been presented with the support of $\left( {{\gamma }, {\gamma }^{\prime} } \right)$-open set, $\left( {{\gamma }, {\gamma }^{\prime} } \right)$-closed set, ${\alpha }_{\left( {{\gamma }, {\gamma }^{\prime} } \right)}$-open set, ${\alpha }_{\left( {{\gamma }, {\gamma }^{\prime} } \right)}$-closed set, $\left( {\gamma ,\gamma^{\prime} } \right)$-regular space, $\left( {\gamma ,\gamma^{\prime} } \right)$-normal space and operators like $cl_{\tau_{\left( {\gamma ,\gamma^{\prime}} \right)} }, int_{\tau_{\left( {\gamma ,\gamma^{\prime}} \right)} }$ $int_{\tau_{\alpha_{\left( {\gamma ,\gamma^{\prime}} \right)} } }, cl_{\tau_{\alpha_{\left( {\gamma ,\gamma^{\prime}} \right)} } }$. Moreover, some of its properties have been premeditated with the aid of bioperation-topological spaces. The generalized open sets and generalized closed sets play a vital role in the study of general topology as well as operation topology. The ${\alpha }_{\left( {{\gamma }, {\gamma }^{\prime} } \right)}$-generalized open sets along with ${\alpha }_{\left( {{\gamma }, {\gamma }^{\prime} } \right)}$-generalized closed sets have been utilized to elaborate various definitions of $((\gamma ,\gamma^{\prime}) ,\alpha_{\left( {\beta ,\beta^{\prime}} \right)} )$-generalized continuous mappings, $((\gamma ,\gamma^{\prime} ),\alpha_{\left( {\beta ,\beta^{\prime}} \right)} )$-generalized open mappings, $((\gamma ,\gamma^{\prime} ),\alpha_{\left( {\beta ,\beta^{\prime}} \right)} )$-closed mappings, $\alpha_{\left( {\gamma ,\gamma^{\prime} } \right)\left( {\beta ,\beta^{\prime}} \right)}$-generalized continuous mapping, $\alpha_{\left( {\gamma ,\gamma^{\prime} } \right)\left( {\beta ,\beta^{\prime}} \right)}$-generalized open mapping and $\alpha_{\left( {\gamma ,\gamma^{\prime} } \right)\left( {\beta ,\beta^{\prime}} \right)}$-generalized closed mapping which are provided so as to study and understand the concepts discussed in the current paper work. The corollaries provide the requirement for $Y_{ToS}$ to be a $\left( {\beta ,\beta^{\prime}} \right)^\alpha$-nor spa. With the aid of the definition of the ultra-nor spa, the concept of ultra- $\left( {\gamma ,\gamma^{\prime} } \right)^{\alpha }$-nor spa definition has been mentioned. Applying the knowledge of ultra-$\left( {\gamma ,\gamma^{\prime} } \right)^{\alpha }$-nor spa definition, certain theorems have been developed. The idea of $((\gamma ,\gamma^{\prime} ),\alpha_{(\beta ,\beta^{\prime})} )$-$g$ -OPM (CLM) and $\alpha_{\left( {\gamma ,\gamma^{\prime} } \right)\left( {\beta ,\beta^{\prime}} \right)}$-$g$ -OPMa (CLMa) is used to discuss various comparative statements as well as theorems.