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?