Quotient topology

The quotient topology on a set $Y$ induced by a surjective function $q:X\to Y$ from a topological space $X$ is defined by declaring a subset $U\subseteq Y$ to be open if and only if $q^{-1}(U)$ is open in $X$.

With this topology, the map $q$ is automatically continuous , and the quotient topology is the finest topology on $Y$ for which this is true. A common special case is when $Y$ is the quotient set $X/\!\sim$ for an equivalence relation $\sim$ on $X$, with $q$ the canonical projection.

Examples:

• Let $X=[0,1]$ (as a subspace of $\mathbb{R}$) and identify $0$ and $1$; the resulting quotient space carries the quotient topology and models a circle.
• If $A\subseteq X$ is collapsed to a single point by a surjection $q:X\to Y$, then $Y$ has the quotient topology determined by this identification map.
• If $q:X\to Y$ is a bijection, the quotient topology on $Y$ agrees with the topology transported by $q$, and $q$ is a homeomorphism exactly when $q^{-1}$ is also continuous.