On Internal Characterizations of Completely L-Regular Spaces

Complete L-regularity is internally characterized in terms of separating chains of open L-sets. A possible characterization in terms of normal and separating families of closed L-sets is discussed and it is shown that spaces admitting such families are completely L-regular. The question of whether the converse holds true remains open. Some partial solutions are however given, e.g. in the class of countably compact spaces.