\[ \DeclareMathOperator{\O}{O} \]

Assume there are two quotient maps \(q_1\colon X\to Y_1\) and \(q_2\colon X\to Y_2\) that make the same identification, i.e. \(q_1(x)=q_1(x')\) if and only \(q_2(x)=q_2(x')\). Then there is a unique homeomorphism \(\varphi\colon Y_1\to Y_2\) such that \(q_2=\varphi\circ q_1\). [1, Theorem 3.75]

Remark

This theorem is useful for showing that a given quotient space is homeomorphic to a known space, e.g.
- \(I/\sim \;\approx \mathbb{S}^1\)
- \(\mathbb{R}^n/\O(n)\approx \mathbb{R}^{\ge 0}\)

References Link to heading

J. Lee, Introduction to Topological Manifolds. New York, NY: Springer New York, 2011. doi:10.1007/978-1-4419-7940-7