Functionally Separation Axioms on General Topology

Abstract

In this paper, we define a new family of separation axioms in the classical topology called functionally $T_i$ spaces for $i=\mathrm{0,1,2}$ . With the assistant of illustrative examples, we reveal the relationships between them as well as their relationship with $T_i$ spaces for $i=\mathrm{0,1,2}$ . We demonstrate that functionally $T_i$ spaces are preserved under product spaces, and they are topological and hereditary properties. Moreover, we show that the class of each one of them represents a transitive relation and obtain some interesting results under some conditions such as discrete and Sierpinski spaces.