Quotient Topology — Mathematics Knowledge System

Overview

The quotient topology is the standard way to build new topological spaces by 'gluing together' points of an existing space according to some equivalence relation. Given a space X and an equivalence relation ~ on X, the quotient set X/~ collects points into equivalence classes, and the quotient map q: X → X/~ sends each point to its class. The quotient topology declares a subset U of X/~ to be open exactly when its preimage q⁻¹(U) is open back in X — this is the unique topology on X/~ making q continuous in the strongest possible sense compatible with the gluing. This construction produces many familiar spaces from simple building blocks: identifying the two endpoints of an interval produces a circle, and identifying antipodal points of a sphere produces the projective plane.

Intuition

Imagine taking a strip of paper and physically gluing two edges together: points that get glued to each other become 'the same point' in the new shape. The quotient topology formalizes this gluing precisely: instead of a physical strip and glue, you specify an equivalence relation (which points are 'the same') on the original space, and a set in the resulting glued-up space counts as open exactly when un-gluing it back in the original space — pulling it back through the quotient map — gives you an open set there. Gluing the two ends of a line segment [0,1] together (identifying 0 with 1) yields a circle, capturing the everyday sense in which 'bending a strip into a loop and taping the ends' is a topological, not just a physical, operation.

Formal Definition

Definition

Let X be a topological space, ~ an equivalence relation on X, and q: X → X/~ the map sending each point to its equivalence class. The quotient topology on X/~ is defined by:

U \subseteq X/{\sim} \text{ is open} \iff q^{-1}(U) \text{ is open in } X

A subset of the quotient set is open precisely when its preimage under q is open in X

Quotient topology

q: X \to X/{\sim}, \quad q(x) = [x] = \{ y \in X : y \sim x \}

q sends each point to its equivalence class; q is automatically continuous and surjective by construction

Quotient map

[0,1] / (0 \sim 1) \ \cong\ S^1

Identifying the two endpoints of a closed interval yields a space homeomorphic to the circle

Circle from an interval

Notation

Notation	Meaning
$\sim$	An equivalence relation on X, specifying which points are to be identified
$[x]$	The equivalence class of x — the set of all points identified with x
$X/{\sim}$	The quotient set of equivalence classes, equipped with the quotient topology
$q: X \to X/{\sim}$	The quotient map sending each point to its equivalence class

Properties

Quotient map is continuous

q: X \to X/{\sim} \text{ is always continuous, by construction of the quotient topology}

Universal property

\text{A function } f: X/{\sim} \to Z \text{ is continuous} \iff f \circ q : X \to Z \text{ is continuous}

Condition: This characterizes the quotient topology uniquely as the finest topology on X/~ making q continuous.

Finest topology making q continuous

\text{The quotient topology is the FINEST topology on } X/{\sim} \text{ for which } q \text{ is continuous.}

Non-Hausdorff quotients

\text{Quotients of Hausdorff spaces need not be Hausdorff.}

Example: Identifying all of ℝ\{0} with a single point but keeping 0 separate can produce a non-Hausdorff 'line with two origins'-type space.

Worked Examples

Every interior point x ∈ (0,1) forms its own singleton equivalence class; the endpoints 0 and 1 are merged into one class.
$[x] = \{x\} \text{ for } x\in(0,1), \quad [0]=[1]=\{0,1\}$
The quotient map wraps the segment around and glues its two ends together, and one can check the resulting space with the quotient topology is homeomorphic to the circle S¹.
$[0,1]/(0\sim1) \cong S^1$

Answer: The quotient is (homeomorphic to) a circle, formed by gluing the two endpoints of the interval together.

Practice Problems

Difficulty 5/10

In the quotient topology on X/~, a set U is defined to be open when:

U is finiteq⁻¹(U) is open in Xq(U) is open in XU contains the equivalence class of every point

Difficulty 6/10

What space results from identifying all points of the closed interval [0,1] to a single point?

Difficulty 7/10

Prove that the quotient map q: X → X/~ is always continuous under the quotient topology.

Quiz

Identifying the two endpoints 0 and 1 of the interval [0,1] via the quotient topology produces a space homeomorphic to:

A pointThe real lineThe circle S¹The square

The quotient topology on X/~ is characterized as:

The coarsest topology making q continuousThe finest topology making q continuousAny topology making q a homeomorphismThe discrete topology on X/~

Summary

Given an equivalence relation ~ on X, the quotient topology on X/~ declares U open iff its preimage q⁻¹(U) under the quotient map is open in X.
This is the finest topology on X/~ making the quotient map q continuous, and it satisfies a universal property: f: X/~ → Z is continuous iff f∘q is continuous.
Gluing constructions like identifying the endpoints of [0,1] to form a circle are formalized precisely by the quotient topology.

References

WebsiteWikipedia — Quotient space (topology)