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Suppose \(X\) is a topological space and \(G\) a group acting on \(X\). We define an equivalence relation on \(X\) by saying \(x_1\sim x_2\) if there is an element \(g\in G\) such that \(g\cdot x_1=x_2\). The corresponding equivalence classes are the orbits of the group action. The resulting quotient space is denoted by \(X/G\), and is called orbit space of the action.

Examples Link to heading

• The orbit space \(\mathbb{R}^n/\O(n)\) is homeomorphic to \(\mathbb{R}^{\ge 0}\).
• The orbit space \(\mathbb{R}^n\setminus \{0\}/\mathbb{R}^*\) is the real projective space \(\mathbb{P}^n\).
• The orbit space of a subgroup action is its coset space.