TITLE: compact quotient
QUESTION [11 upvotes]: Let X be a topological space that is not too bad (let's say "not too bad" = "compactly generated Hausdorff"), and let ∼ be an equivalence relation such that X /∼ is compact Hausdorff.
Does there exist a compact subspace A⊂X that meets every equivalence class of ∼? (This would then imply that A /∼ is homeomorphic to X /∼).

REPLY [5 votes]: Here is a counterexample, inspired by Henrik Rüping's answer, but somewhat simpler:
Take $X$ to be the disjoint union of $\quad X_1:=([0,1]\times [0,1])\setminus \{(0,0)\}$
and $\quad X_2:=(]0,1]\times [0,1])\cup ([-1,0]\times\{0\})$. Both $X_1$ and $X_2$ are given the subspace topology from $\mathbb R^2$.
The quotient space is $[0,1]^2$, with quotient map $\pi:X\to [0,1]^2$.
The map $\pi|_{X_1}$ is the inclusion $X_1\hookrightarrow [0,1]^2$.
The map $\pi|_{X_2}$ is a continuous bijection $X_2 \to [0,1]^2$ given by $\pi(-a,0)= (0,a)$ for $(-a,0)\in [-1,0]\times\{0\}$.