Quotient Topology

Overview

A quotient topology describes a space formed by identifying points. It appears when an interval is bent into a circle, when opposite edges of a square produce a torus, and when symmetry orbits replace individual points. The equivalence classes specify the underlying set, but the topology must still be supplied. The quotient construction supplies it through inverse images.

Prerequisites

Understand equivalence relations, surjective maps, and inverse-image continuity. A subset of the original space is saturated if it is a union of entire equivalence classes. Saturation is useful for recognizing which open subsets can descend to the quotient.

Definition and conventions

Let $q:X\to Y$ be a surjection from a topological space onto a set. The quotient topology on $Y$ is defined by

It is the finest topology making $q$ continuous. For an equivalence relation, take $Y=X/{\sim}$ and $q(x)=[x]$.

A quotient map is a surjection between already topologized spaces satisfying this equivalence. Mere continuity gives only one implication. Quotients are not assumed Hausdorff, second countable, or manifolds.

Intuition

An identified point represents all members of one class. A neighborhood of that point must pull back to a neighborhood of every representative. The quotient topology encodes this simultaneous requirement. It can preserve familiar geometry, but identifying a badly chosen subset can also make distinct classes impossible to separate.

Examples and non-examples

Identify $0$ and $1$ in $[0,1]$. The function $t\mapsto e^{2\pi it}$ is constant exactly on the identified class and otherwise injective. It induces a continuous bijection from the quotient to $S^1$. Since the quotient is a continuous image of a compact space and the circle is Hausdorff, the induced map is a homeomorphism.

For a contrasting example, identify all rational points of $\mathbb R$ to one class and leave irrational points separate. A saturated open neighborhood of an irrational point contains a rational point and hence every rational point. It cannot be disjoint from any neighborhood of the rational class. The quotient is therefore not Hausdorff.

Important properties

A map $f:Y\to Z$ is continuous exactly when $f\circ q$ is continuous. One proves this by comparing inverse images of every open subset of $Z$. This universal property often makes an explicit description of all quotient open sets unnecessary.

Surjective continuous open maps and surjective continuous closed maps are quotient maps. Continuity is essential: the identity from the usual real line to the discrete real line is surjective, open, and closed, but not continuous and hence not quotient. In particular, a continuous surjection from a compact space to a Hausdorff space is closed and therefore quotient. Neither openness nor closedness is required of every quotient map.

Connections and further reading

Quotients build cell complexes, projective spaces, and orbit spaces. Smooth or symplectic reduction needs extra hypotheses beyond this topological construction. Read Petrunin, Claim 6.17 in Chapter 6E, Chapter 6F, and Chapter 9D for the inspected open/closed-map criterion, quotient definition, and compact-to-Hausdorff result.