Quotient topology

Quotient topology is a concept in the branch of mathematics known as topology.

Definition

Let $X$ be a topological space, and $A$ a set, and let $p\colon X\to A$ be a surjection. The quotient topology on $A$ induced by $p$ is the topology whose open sets are the sets $U\subseteq A$ such that $p^{-1}(U)$ is an open set in $X$.

Examples

Quotient topologies can often be visualized as gluing elements of a topological space together.

Let $X=[0,1]$ with the usual topology (as a subspace of the reals), $A=[0,1)$, and $p\colon X\to A$ be given by $p(1)=0$ and $p(x)=x$ for $x\neq 1$. Then $A$ under the quotient topology is homeomorphic to the circle. Indeed, we can visualize what happened as a gluing operation: the two endpoints of the interval were glued together to create a closed loop.