Abstract:
Given a map on a set we examine under what conditions there is a separable metrizable or an hereditarily Lindelöf or a Lindelöf topology on with respect to which is a continuous map. For separable metrizable and hereditarily Lindelöf, it turns out that there is such a topology precisely when the cardinality of is no greater than , the cardinality of the continuum. We go on to prove that there is a Lindelöf topology on with respect to which is continuous if either or for some , where and for any ordinal and limit ordinal .