CONTINUITY IN SEPARABLE METRIZABLE AND LINDELÖF SPACES

Abstract. Given a map T: X → X on a set X we examine under what con-ditions there is a separable metrizable or an hereditarily Lindelöf or a Lindelöf topology on X with respect to which T is a continuous map. For separable metrizable and hereditarily Lindelöf, it turns out that there is a such a topology precisely when the cardinality ofX is no greater than the cardinality of the con-tinuum. We go onto prove that there is a Lindelöf topology on X with respect to which T is continuous, if either Tc (X) = Tc ++1(X) 6 = ∅ or Tα(X) = ∅ for some α < c+, where Tα+1(X) =