Suppose \(X\) is a topological space , \(Y\) a set and \(q\colon X\to Y\) a surjective map . We declare a subset \(U\subseteq Y\) open if and only \(q^{-1}(U)\) is open in \(X\). This is called the quotient topology on \(Y\) induced by the map \(q\).
Suppose there is a partition for \(X\) defined by an equivalence relation . Then \(X/\sim\) endowed with the quotient topology (where \(q\colon X\to X/\sim\) is the natural projection) is called quotient space of \(X\).
Remarks
(Not) Inherited properties Link to heading
- Locally Euclidean quotient spaces of second countable spaces are second countable.
- Quotient spaces of connected spaces are connected.
- A quotient space of a Hausdorff space is not Hausdorff in general. For example, the topology of \(\mathbb{R}/\mathbb{Q}\) is the trivial one.