If $(X,\tau)$ is a topological space, we say $Y\subseteq X$ is *homeomorphism-fixing* if the only homeomorphism $\varphi:X\to X$ such that for all $y\in Y$ we have $\varphi(y)=y$ is the identity map. Moreover we say that $Y$ is *minimally homeomorphism-fixing* if for all $y\in Y$ the set $Y\setminus \{y\}$ is not homemorphism-fixing.


What is an example of an infinite Hausdorff space without minimally homeomorphism-fixing subsets?



Homeomorphism-fixing subsets
The real numbers.


Any dense subset, like the rationals is homeomorphism fixing. Conversely if that subset $Y$ is not dense, we can find a point $x\in X$ and a small interval around $x$ that does not contain any element of $Y$. Now it is easy to construct a homeomorphism that is the identity outside of that interval (and thus fixes $Y$) and does something on that interval (reparametrizing).


If you remove a single point from a dense subset, that subset is still dense and thus homeomorphism fixing. Thus there are no minimal homeomorphism fixing subsets.


