Let $(X,\tau)$ be a topological space. We define the "moving" relation by setting $$ x \simeq\_m y \text{ iff there is a homemomorphism }\varphi: X\to X \text{ such that } \varphi(x) = y.$$


Clearly $\simeq\_m$ is an equivalence relation. We call a space "immovable" if $\simeq\_m$ is the diagonal $\Delta\_X=\{(x,x):x\in X\}$.


If $(X,\tau)$ is a topological space, is $X/\simeq\_m$ always immovable?



"Immovable" topological spaces
No. Let $X$ be the disjoint union of the real line and one isolated point. The quotient by "movability" collapses the real line to one point, so the quotient is a discrete space of two points, which is not immovable.


