If $(X,\tau)$ is a topological space and $x,y\in X$ we say that $x$ is mappable to $y$ if there is a non-constant map $f:X\to X$ with $f(x) = y$.

Is there a Hausdorff space $X$ with more than one point such that whenever $x\neq y\in X$ and $x$ is mappable to $y$, then $y$ is not mappable to $x$?

If $(X,\tau)$ is a topological space and $x,y\in X$ we say that $x$ is mappable to $y$ if there is a non-constant continuous map $f:X\to X$ with $f(x) = y$.

Is there a Hausdorff space $X$ with more than one point such that whenever $x\neq y\in X$ and $x$ is mappable to $y$, then $y$ is not mappable to $x$?