Let $G$ be a compact (Hausdorff) group acting jointly continuously on a compact (Hausdorff) space $X$. Given $x \neq y \in X$, does there exist a continuous $G$-invariant pseudometric on $X$ such that $d(x,y) > 0$?
That is, given $x \neq y$, I want to find (the existence of) a continuous function $d : X^2 \to \mathbb{R^{\geq 0}}$ such that
- For all $x$, $d(x,x) = 0$.
- For all $x,y$, $d(x,y) = d(y,x)$.
- For all $x,y,z$, $d(x,z) \leq d(x,y) + d(y,z)$.
- For all $x,y,g$, $d(gx,gy) = d(x,y)$.
- $d(x,y) > 0$ for the specific $x$ and $y$ we're given.
We may not assume either $G$ or $X$ is separable.
The context is that I'm a graduate student in mathematics working on a research problem where such an invariant pseudometric would be useful, and finding one is a little bit outside my usual expertise. I'll attempt to build one (and post the construction here if I do), but ideally I'd like to cite this as an easy consequence of some well-known, textbook theorem.
The positive answer to your question (even for an arbitrary Tychonoff space $X$) follows from Theorem A from the paper “Invariant pseudometrics on Palais proper $G$-spaces’’ by Antonyan and de Neymet.