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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?

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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 $\omega_1$.

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    $\begingroup$ Or more simply, just take $\omega+1$. $\endgroup$ – Eric Wofsey Aug 12 '15 at 8:06

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