How do open subsets of X/G look like?

Question

Let $G$ act continuously on $X$, where $X$ is a topological space.

So I wonder about how open subsets look like in $X/G$.

The action $a$ is defined as $a(g,x)=g.x$.

Answer 1

The quotient space topology in the space of orbits $X/G$ is induced by the topology in $X$ via the projection $\pi \colon X \to X/G$ where $\pi(x) = Gx$. More specifically a subset $U \subset X/G$ is open if and only if $\pi^{-1} U$ is open.

Another way of saying this is that a set $\mathcal{O}$ of orbits is open if and only if $\bigcup \mathcal{O}$ is open in $X$.

Answer 2

The open sets of $X/G$ are precisely images of $G$-invariant open subsets of $X$, or if you want, images of saturated open subsets of $X$. This works for any quotient topology where the map $f\colon X \to Y$ is surjective. Note however that in the case of spaces of orbits $X/G$ the map $X\to X/G$ is moreover open, that is, the image of an open subset of $X$ is open, equivalently, the saturation of an open subset $U$ of $X$, being $\cup_{g\in G} g\cdot U$, is also open.

Equivalently, you can think in terms of closed subsets: the closed ones of $X/G$ are images of saturated($G$-invariant) closed subsets of $G$. In this case, the map $X\to X/G$ may not be closed, but it surely is if say $G$ is finite.