The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection

The quotient topology. Let $(X,\mathcal{T})$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), let $Y$ be a set and let $q:X\to Y$ be a surjection (Injection, surjection, bijection). The quotient topology on $Y$ induced by $q$ is the final topology of the one-element family $(q)$ (The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology):

${\mathcal{T}}_q:=\{V\subseteq Y:q^{-1}[V]\in \mathcal{T}\}\mathrm{.}$

That this is a topology is discharged in The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology, where every final topology is verified to satisfy (T1), (T2) and (T3). Dually, $C\subseteq Y$ is closed in ${\mathcal{T}}_q$ exactly when $q^{-1}[C]$ is closed in $X$, because $q^{-1}[Y\setminus V]=X\setminus q^{-1}[V]$.

Quotient map. A surjection $q:X\to Y$ between topological spaces is a quotient map (also identification map) when the topology of $Y$ is the quotient topology of $q$, that is when

$V\text{ is open in }Y⟺q^{-1}[V]\text{ is open in }X(V\subseteq Y)\mathrm{.}$

The implication from left to right is exactly continuity of $q$ (Continuity of a map of topological spaces at a point and globally, For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f(\bar{A})\subseteq \overline{f(A)}$ clause (b)), so a quotient map is a continuous surjection with the extra property that its topology is as fine as continuity permits. Equivalently, and this form is used as often: $C$ is closed in $Y$ if and only if $q^{-1}[C]$ is closed in $X$.

Saturated sets. Let $q:X\to Y$ be a surjection. A subset $A\subseteq X$ is saturated with respect to $q$ when

$A=q^{-1}[q[A]],$

equivalently when $A$ is a union of fibres $q^{-1}[\{y\}]$, equivalently when $x\in A$ and $q(x^{\mathrm{\prime }})=q(x)$ imply $x^{\mathrm{\prime }}\in A$. The set $q^{-1}[q[A]]$ is the saturation of $A$, and it is the smallest saturated set containing $A$. The map $V\mapsto q^{-1}[V]$ is a bijection from $\mathcal{P}(Y)$ onto the saturated subsets of $X$, with inverse $A\mapsto q[A]$, since $q$ is surjective; under it the open sets of the quotient topology correspond exactly to the saturated open subsets of $X$. That correspondence is the working description of the quotient topology: to know the open sets of $Y$ is to know which open subsets of $X$ are saturated.

Quotient by an equivalence relation. Let $\sim$ be an equivalence relation on $X$, that is a relation that is reflexive ($x\sim x$), symmetric ($x\sim x^{\mathrm{\prime }}$ implies $x^{\mathrm{\prime }}\sim x$) and transitive ($x\sim x^{\mathrm{\prime }}$ and $x^{\mathrm{\prime }}\sim x^{\mathrm{\prime }\mathrm{\prime }}$ imply $x\sim x^{\mathrm{\prime }\mathrm{\prime }}$). Write $[x]:=\{x^{\mathrm{\prime }}\in X:x^{\mathrm{\prime }}\sim x\}$ for the equivalence class of $x$; distinct classes are disjoint and their union is $X$. The quotient set is $X\mathrm{/}\text{\hspace{0.17em}⁣}\sim :=\{[x]:x\in X\}$ and the canonical projection is

$\pi :X\to X\mathrm{/}\text{\hspace{0.17em}⁣}\sim ,\pi (x):=[x],$

which is a surjection by construction. The set $X\mathrm{/}\text{\hspace{0.17em}⁣}\sim$ always carries the quotient topology of $\pi$, and $(X\mathrm{/}\text{\hspace{0.17em}⁣}\sim ,{\mathcal{T}}_{\pi })$ is called an identification space. A subset of $X$ is saturated for $\pi$ exactly when it is a union of equivalence classes.

A convenient special case: collapsing a subset. For $\mathrm{\varnothing }\mathrm{\ne }B\subseteq X$ let ${\sim }_B$ be the relation whose classes are $B$ itself and the singletons $\{x\}$ for $x\mathrm{\notin }B$; this is an equivalence relation, its classes being a partition of $X$. The resulting quotient is written $X\mathrm{/}B$, and its canonical projection is a surjection identifying all of $B$ to a single point and doing nothing else. Saturated sets for ${\sim }_B$ are the sets $A$ with $A\cap B\in \{\mathrm{\varnothing },B\}$.

Two conventions used throughout. First, "quotient map" is a property of a map together with the two topologies, never of the map alone. Second, a quotient topology is determined by $q$ together with the topology on its domain, and not by the underlying pair of sets $(X,Y)$: two different surjections onto the same set can give different topologies, the same surjection gives different topologies when its domain is retopologised, and where more than one is in play the map is named.

• The quotient construction on sets is the one this library has already used. The integers (The integers as equivalence classes of pairs of naturals) and the rationals (The rationals as equivalence classes of pairs of integers) are quotient sets of exactly the shape above, and the only new content here is the topology carried along by $\pi$.

• Why saturation is the right notion. An open $U\subseteq X$ that is not saturated has an image $q[U]$ whose preimage is strictly larger than $U$, and nothing forces that preimage to be open; so $q[U]$ need not be open, and indeed a quotient map need not be an open map. That failure is recorded on this page as a false statement and is not a defect of the construction: the quotient topology is defined by preimages precisely because images behave badly.

• The quotient topology is the finest making $q$ continuous. That is the general fact about final topologies from Characteristic properties: a map into a space with the initial topology is continuous iff every composite with the defining family is, a map out of a space with the final topology is continuous iff every composite with the defining family is, and the two topologies are respectively the coarsest and the finest making that family continuous, read for a one-element family, and it is what makes "as many open sets as continuity permits" precise.