Continuity in separable metrizable and Lindelöf spaces

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 $\lambda$.