Separation axioms on function spaces defined on bitopological spaces

Abstract

In this paper, we generalize separation axioms to the function space \(p\) − \(C_\omega(Y,Z)\) and study how they relate to separation axioms defined on the spaces \((Z, \delta_i)\) for \(i = 1, 2,\)\( (Z, \delta_1, \delta_2)\), 1 − \(C_\xi(Y,Z)\) and 2 −\(C_\xi(Y,Z)\). We show that the space \(p\) −\(C_\omega(Y,Z)\) is \(pT_◦\), \(pT_1\), \(pT_2\) and pregular, if the spaces \((Z, \delta_1)\) and \((Z, \delta_2)\) are both \(T_◦\), \(T_1\), \(T_2\) and regular respectively. The space \(p\)−\(C_\omega(Y,Z)\) is also shown to be \(pT_◦\), \(pT_1\), \(pT_2\) and pregular, if the space \((Z, \delta_1, \delta_2)\) is \(p\) −\(T_◦\), \(p\) −\(T_1\), \(p\) −\(T_2\) and \(p\)-regular respectively. Finally, the space \(p\) −\(C_\omega(Y,Z)\) is shown to be \(pT_◦\), \(pT_1\), \(pT_2\) and pregular, if and only if the spaces 1 −\(C_\xi(Y,Z)\) and 2 −\(C_\xi(Y,Z)\) are both \(T_0\), \(T_1\), \(T_2\), and only if the spaces 1 −\(C_\xi(Y,Z)\) and 2 −\(C_\xi(Y,Z)\) are both regular respectively.