A study of some spaces related to {\}-open sets

The purpose of this paper is to introduce weakly ${}-{R0}$ spaces, ${}-US$ spaces and to obtain their basic properties. It is shown that weakly ${}-{R0}$ and }-US spaces are preserved under pre-${}$-closed injection and pre-{}$-closed bijection respectively

Operation approach to ß-open sets and applications

In this paper, we introduce the concept of an operation $gamma$ on a family of $beta$-open sets denoted by $beta O(X)$ in a topological space $(X,tau)$. Using the operation $gamma$ on $beta O(X)$, we introduce the concept of $beta$-$gamma$-open sets, and investigate the related topological properties. We also introduce the notion of $beta$-$gamma$-$T_{i}$ spaces $(i =0, 1/2, 1, 2)$ and study some topological properties on them. Further, we introduce $beta$-$(gamma, b)$-continuous maps and investigate basic properties. Finally, we investigate a general operation approach to $beta$-closed graphs of mappings

More on Diophantine Equations

In this paper, we will find the solutions of many Diophantine equations.Some are of the form 2(3 x )+ 5(7y ) +11=z2 for non negative x,y and z. we also investigate solutions ofthe Diophantine equation 2(x+3) +11(3y ) ─ 1= z2 for non negative x,y and z and finally, westudy the Diophantine equations (k!×k)n = (n!×n)k and ( 𝒌! 𝒌 ) 𝒏 = ( 𝒏! 𝒏 ) 𝒌 where k and n are positive integers. We show that the first one holds if and only if k=n and the second one holds if and only if k=n or (k,n) =(1,2),(2,1).We also investigate Diophantine equation u! + v! = uv and u! ─ v! = uv .</jats:p