Quotient Space (Topology)

A quotient space is a topological space constructed by treating points related by an equivalence relation as a single point. Its underlying set consists of the equivalence classes, while its topology is determined by the topology of the original space. This gives a precise meaning to geometric operations such as gluing edges together or collapsing a subset to a point. The construction belongs to topology and is fundamental to the study of spaces assembled by identification. (pi.math.cornell.edu)

Definition

Let $X$ be a topological space and let $\sim$ be an equivalence relation on $X$ . The quotient set is

X/{\sim}=\{[x]:x\in X\},

where the equivalence class of $x$ is

[x]=\{y\in X:y\sim x\}.

The canonical projection

q:X\longrightarrow X/{\sim},\qquad q(x)=[x],

is a surjective function. The quotient topology is defined by

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

Thus openness in the quotient is tested by pulling a set back to the original space. (maths.dur.ac.uk)

This definition produces a topology because inverse images preserve arbitrary unions and finite intersections, and the inverse images of the empty set and the whole quotient are empty and $X$ , respectively. It is the finest topology—the one with the most open sets—for which $q$ is a continuous map. Any topology making $q$ continuous can contain only sets whose inverse images are open. (pi.math.cornell.edu)

Equivalently,

F\subseteq X/{\sim}\text{ is closed} \quad\Longleftrightarrow\quad q^{-1}(F)\text{ is closed in }X.

Here “closed” refers to a closed set in the appropriate space. (math.ucla.edu)

Quotient maps and saturated sets

More generally, a surjection $p:X\to Y$ between topological spaces is a quotient map if

U\text{ is open in }Y \quad\Longleftrightarrow\quad p^{-1}(U)\text{ is open in }X.

Every such map realizes $Y$ as the quotient obtained by identifying points with the same image. Continuity alone supplies only one direction of this equivalence, so a continuous surjection need not be a quotient map. (pi.math.cornell.edu)

A subset $A\subseteq X$ is saturated if it is a union of entire equivalence classes, or equivalently,

q^{-1}(q(A))=A.

The definition shows that open sets in the quotient correspond exactly to saturated open subsets of $X$ . An arbitrary open subset of $X$ need not have open image: its saturation may fail to be open. This explains why quotient maps need not be open maps. (jde27.uk)

Useful sufficient conditions are:

A continuous, surjective open map is a quotient map.
A continuous, surjective closed map is a quotient map.
A continuous surjection from a compact space to a Hausdorff space is a quotient map.

For the last condition, closed subsets of the domain are compact, and their images are compact and therefore closed in the Hausdorff codomain. (pi.math.cornell.edu)

Universal property

The central universal property concerns maps out of a quotient. Suppose $f:X\to Z$ is continuous and constant on equivalence classes:

x\sim y\implies f(x)=f(y).

Then there is a unique continuous map

\bar f:X/{\sim}\longrightarrow Z

such that

f=\bar f\circ q, \qquad \bar f([x])=f(x).

Constancy on classes makes this formula well-defined, and surjectivity of $q$ gives uniqueness. (math.toronto.edu)

Continuity follows from the identity

q^{-1}\bigl(\bar f^{-1}(V)\bigr)=f^{-1}(V)

for every open $V\subseteq Z$ . More generally, a function $g:X/{\sim}\to Z$ is continuous if and only if $g\circ q$ is continuous. This is often called the descending-map property: a continuous construction on $X$ descends to the quotient precisely when it respects the identifications. (ma.imperial.ac.uk)

Geometric examples

A circle from an interval

Identify the endpoints of $[0,1]$ , leaving every interior point in its own class. The resulting space is homeomorphic to the circle $S^1$ . The map

t\longmapsto e^{2\pi i t}

identifies exactly $0$ and $1$ , and is a quotient map because its domain is compact and its codomain is Hausdorff. Notice that this map is not open: a sufficiently short relative neighborhood of $0$ maps to an arc that contains no full circle-neighborhood of the identified point. (ma.imperial.ac.uk)

Surfaces from a square

Starting with $[0,1]^2$ , impose

(0,t)\sim(1,t),\qquad (s,0)\sim(s,1).

The quotient is a torus, homeomorphic to $S^1\times S^1$ . These identifications also place all four corners in one class. Gluing just the left and right edges with reversed parameter,

(0,t)\sim(1,1-t),

instead produces a Möbius strip. Reversing one pair of edge identifications while also gluing the other pair produces a Klein bottle. The equivalence relation includes all identifications forced by transitivity, not merely the explicitly listed pairs. (math.ucla.edu)

Collapsing a subspace

For a nonempty subset $A\subseteq X$ , the notation $X/A$ usually means that all of $A$ is identified to one point and every point outside $A$ remains distinct. A standard example is

D^n/S^{n-1}\cong S^n:

collapsing the boundary of a closed $n$ -ball gives an $n$ -sphere. This is a topological identification, not deletion of the boundary. (math.ucla.edu)

Orbit spaces

A group action on $X$ defines an equivalence relation by declaring two points equivalent when they belong to the same orbit. The resulting quotient $X/G$ is an orbit space. For example, integer translations of the real line give

\mathbb R/\mathbb Z\cong S^1.

Here $\mathbb R/\mathbb Z$ carries the quotient topology, not merely the structure of a set of cosets. (jde27.uk)

When each group element acts by a homeomorphism, the orbit projection is open. Indeed, for open $U\subseteq X$ ,

q^{-1}(q(U))=\bigcup_{g\in G}g(U),

which is open. This additional property distinguishes orbit projections from general quotient maps. Quotients by suitable actions also provide constructions of manifolds, although an arbitrary action need not produce a manifold. (jde27.uk)

Preservation and failure of topological properties

Compactness, connectedness, and path connectedness pass from $X$ to its quotient, because the projection is a continuous surjection. The converses do not hold: any nonempty space can be collapsed to a single point. (pmelvin.blogs.brynmawr.edu)

Separation properties require greater care. The quotient is $T_1$ , meaning that every singleton is closed, exactly when every equivalence class is closed in $X$ . This follows by applying the closed-set criterion to $\{[x]\}$ . Closed classes alone, however, do not ensure that the quotient is Hausdorff. (math.ucla.edu)

A standard counterexample is the line with two origins. Take two disjoint copies of $\mathbb R$ and identify corresponding nonzero points, but keep the two origins distinct. Every class is closed. Nevertheless, any neighborhoods of the two origins intersect at nearby nonzero points, so the quotient is not Hausdorff, even though the original disjoint union is Hausdorff. Consequently, even a quotient of a metric space need not be metrizable. (ma.imperial.ac.uk)

For a compact Hausdorff space $X$ , there is a stronger criterion. Define

R=\{(x,y)\in X\times X:x\sim y\},

where $X\times X$ has the product topology. Then

X/{\sim}\text{ is Hausdorff} \quad\Longleftrightarrow\quad R\text{ is closed in }X\times X.

Under these conditions the quotient projection is also closed. Closedness of the entire relation is stronger than closedness of each individual class. (paperman.name)

Role in algebraic topology

In algebraic topology, quotient constructions express spaces in terms of pieces and attaching instructions. A CW complex is built by attaching disks along their boundaries. If $A$ is a subcomplex of a CW complex $X$ , collapsing $A$ gives a natural CW structure on $X/A$ : the cells outside $A$ remain, while $A$ becomes one new vertex. (pi.math.cornell.edu)

Other standard constructions include the cone

CX=(X\times[0,1])/(X\times\{0\}),

and the suspension, obtained by collapsing $X\times\{0\}$ and $X\times\{1\}$ to two separate points. In particular, suspending $S^n$ gives $S^{n+1}$ . These constructions depend on the quotient topology: the set of identified points alone does not specify the continuity or neighborhood structure of the resulting space. (pi.math.cornell.edu)

References

Topology Notespi.math.cornell.edu
MAT 327: Introduction to Topologymath.toronto.edu
Point-Set Topology — Chapter 2 Continuous functionsmath.ucla.edu
Algebraic Topology Lecture Notesma.imperial.ac.uk
Topology III Michaelmas 2020maths.dur.ac.uk
03 Quotient topology: group actionsjde27.uk
Math 396. Quotients by group actionsmath.stanford.edu
Topology Lecture Notespmelvin.blogs.brynmawr.edu
The q-Theory of Finite Semigroupspaperman.name
Algebraic Topologypi.math.cornell.edu