Compact Lie group action on non-Hausdorff (but CGWH) space with Hausdorff quotient

Assume that we are in the following situation: a compact Lie group $G$ acts on a compact space $X$ which is not necessarily Hausdorff. $X$ is assumed to be compactly generated and weakly Hausdorff, though. The quotient space $X/G$ is Hausdorff.

Does this imply that $X$ is Hausdorff? I do not think this is true but I'm also unable to find a counterexample since I cannot come up with a non-Hausdorff space that admits a non-trivial $G$-action in the first place.

The reason I'm thinking about this is the following: If the $G$-action on $X$ was additionally free (and $X$ was Hausdorff), then $X$, being compact and Hausdorff, would be completely regular, and hence the quotient map $X \to X/G$ would be a fiber bundle by a result of Bredon. I'm also interested in whether $X \to X/G$ is a fiber bundle if $X,G,X/G$ are as above and the $G$-action is free.

• $X$ weakly Hausdorff: all images of compact Hausdorff spaces in $X$ are closed (implies $T_1$: points are closed). Compactly generated: en.wikipedia.org/wiki/Compactly_generated_space
– YCor
Dec 26, 2016 at 3:16
• One reason it may be hard to think of an example by hand is because you assume the quotient is Hausdorff, and then compactness of $G$ implies that each orbit subspace must be Hausdorff. With a Hausdorff "base" and Hausdorff "fibers," non-Hausdorffness in the "total space" has to come from something weird. Dec 26, 2016 at 16:38
• @GabrielC.Drummond-Cole: Thanks, I had not realized that the weak Hausdorff property on $X$ is sufficient to have Hausdorff orbits until now. I agree that this makes a potential example even less likely to be found. Jan 5, 2017 at 0:08
• It seems that a weakly Haudorff first-countable space is Hausdorff, so the answer is affirmative for first-countable spaces $X$. By the way, what is the answer for finite acting group $G$? Aug 23, 2017 at 10:21
• @TarasBanakh why does WH + first-countable imply Hausdorff? I wasn't able to find this statement, maybe you're confusing this with the fact that every first-countable space is compactly generated (see Strickland's notes, for example). Aug 24, 2017 at 16:36

How about $X=\{ a,b,x,y\}$ with nontrivial open sets $\{a,b\}$ and $\{ x,y\}$ and $G= Z/2$, discrete. The action exchanges $a$ with $b$ and $x$ with $y$.
• Thank you for your comment. Unfortunately, the space $X$ is not weak Hausdorff. If we endow $K := \{c,d\}$ with the discrete topology, $K$ is compact Hausdorff. However, the map $K \to X$ given by $c \mapsto a, d \mapsto x$ is continuous while its image $\{a,x\}$ is not closed in $X$. Nov 28, 2016 at 18:57