If $(X,\tau)$ is a topological space, then we say $A\subseteq X$ is a *fiber* if there is $f:X\to X$ continuous and $y\in X$ such that $A = f^{-1}(\{y\})$. For any $T\_1$-space it is clear that fibers are closed.


In $\mathbb{R}$ the converse holds: all closed sets are fibers. 


**Question.** Is there a connected $T\_2$-space $(X,\tau)$ with $|X|>1$ such that there is a closed subset $A\subseteq X$ that is not a fiber?



Connected $T_2$-space such that not all closed subsets are fibers
Yes. See [Kannan and Rajagopalan - Constructions and applications of rigid spaces, I](https://www.sciencedirect.com/science/article/pii/0001870878900063) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=501093)), particularly their construction 2.2.4, which gives a strongly rigid connected Hausdorff space $Y$. In particular, two-point subsets cannot be fibers. 


