Continuity in separable metrizable and Lindelöf spaces

Abstract:

Given a map $T:X\to X$ on a set $X$ we examine under what conditions 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 such a topology precisely when the cardinality of $X$ is no greater than $\mathfrak {c}$, the cardinality of the continuum. We go on to prove that there is a Lindelöf topology on $X$ with respect to which $T$ is continuous if either $T^{\mathfrak {c}^+}(X)=T^{\mathfrak {c}^++1}(X)\neq \emptyset$ or $T^\alpha (X)=\emptyset$ for some $\alpha <\mathfrak {c}^+$, where $T^{\alpha +1}(X)=T\big (T^\alpha (X)\big )$ and $T^\lambda (X)=\bigcap _{\alpha <\lambda }T^{\alpha }(X)$ for any ordinal $\alpha$ and limit ordinal $\lambda$.