Suppose some group, G, acts on a space, X. Then an orbit of some $x\in X$ is defined as $$G.x = \lbrace g.x \mid g\in G\rbrace$$ Now consider the orbit space, $X/G$, the set of all orbits. I'm finding it a little confusing, to interpret this space. I'll try to formulate my thoughts of this space, and the problem should be somewhat clear:
When thinking of the set of all orbits, what comes to mind for instance in $\mathbb{R}^2$, is loads of lines/curves. However if I consider $X/G$, where each orbit of $X$ is an equivalence class, is each orbit not represented by a point?
If an example is necessary, I can provide one, but I want to keep this quite general for now, as this is part of an assignment.