Continuity in separable metrizable and Lindelöf spaces

Given a map T: X -> X on a set X we examine under what conditions there is a separable metrizable or an hereditarily Lindelof or a Lindelof topology on X with respect to which T is a continuous map. For separable metrizable and hereditarily Lindelof, it turns out that there is such a topology precisely when the cardinality of X is no greater than c, the cardinality of the continuum. We go on to prove that there is a Lindelof topology on X with respect to which T is continuous if either Tc+ (X) = Tc+ + 1 (X) not equal theta or T-alpha(X) = theta for some alpha <c(+), where T alpha+1(X) = T(T-alpha(X)) and T-lambda(X)= boolean AND(alpha