Orbit space

Let \(G\) be a group acting on a topological space \(X\). For \(x,y\in X\) write \(x\sim y\) if there exists \(g\in G\) with \(g\cdot x=y\); this is an equivalence relation whose classes are the orbits of the action. Let \(X/G\coloneqq X/{\sim}\) be the set of orbits and \(q\colon X\to X/G\) the map sending \(x\) to its orbit. The orbit space is \(X/G\) with the quotient topology: \(U\subseteq X/G\) is open if and only if \(\inverseof{q}(U)\) is open in \(X\), so that \(q\) is a quotient map.