This question is a special case of Dominic van der Zypen's question Reconstructing relations with the image relation of a topology, as discussed in the comments, particularly the comment of Eric Wofsey, which explains that if the answer to this question is affirmative, then the answer to Dominic's question is also affirmative.

**Question.** Does every set $X$ have a topology for which the only continuous surjection $f:X\to X$ is the identity map?

My answer to Dominic's question shows that for finite sets, the answer is affirmative, since one need only place an order on $X$, and then let the topology be the up-sets of the order. Every continuous surjection is a permutation of $X$ and order-preserving, and hence the identity. Although it is tempting to try to use the same method with well-orders on an infinite set, it doesn't quite work out in that generality, because the predecessor function on the finite height elements (otherwise fixed) will be a continuous surjection, but not the identity.

For an extreme negative answer, I would ask: can one show that there is no topology on a countably infinite set for which the only continuous surjection is the identity map?