CONTINUITY IN SEPARABLE METRIZABLE AND LINDELOF 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 T(c+) (X) = T(c+) + 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<lambda) T(alpha)(X) for any ordinal alpha and limit ordinal lambda.