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?