For any topological space $(X,\tau)$ we define $$R\_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$
Clearly, $R\_{im}(X,\tau)$ is reflexive, and transitivity follows from the fact that the composition of two continuous surjective maps is continuous and surjective.


Is there a Hausdorff space $(X,\tau)$ such that $R\_{im}(X,\tau)$ is not symmetric?



Hausdorff spaces with asymmetric image relation
There are many such spaces. 


For example $\omega\_1+1= [0,\omega\_1]$ with 
the order topology. A continuous map $f$ with $f(\omega\_1)=0$ must be eventually constant, hence can have only countably many values. 


But there is a continuous surjective selfmap $g$ mapping $0$ to $\omega\_1$. 


