S-continuous functions and certain weak forms of regularity and complete regularity

A topological space is called s-regular if each closed connected set and a point outside it are separated by disjoint open sets. Similarly notion of complete s-regularity is introduced; basic properties of s-regular spaces and completely s-regular spaces are studied and interrelations between them and the standard separation axioms are observed. It is shown that in the class of semilocally connected spaces s-regularity coincides with regularity and complete s-regularity coincides with complete regularity. Moreover, properties of s-continuous functions are studied and it is shown that s-regularity and completely s-regularity are preserved under certain s-continuous mappings.