Let $X,d$ be Hausdorff and let the level sets of a function $f:X\to X$ converge to their image - in other words, $x$ is a limit point of the set $f^{-1}(\{x\})$.
Is the directed graph of the orbit of $f$ in $X$ graph-connected if and only if $X,d$ is connected?
Seems reasonable. I can't find a counterexample. But I'm a novice at topology.
My intuition seems to say that, given the property in this question's first sentence, then what I really need to do is to make a topology on $X$ which says that a subset $Y$ is closed if and only if forall $x\in Y$, $Y$ is closed if and only if $f(x)$ is also in $Y$.
But I have no idea whether that coincides with $f:X,d\to X,d$ being function on a Hausdorff topology having a graph-connected orbit.