nLab quotient space

This entry is about the concept in topology. For quotient vector spaces in linear algebras see there.

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Idea

A quotient space is a quotient object in some category of spaces, such as Top (of topological spaces), or Loc (of locales), etc.

Often the construction is used for the quotient $X/A$ by a subspace $A \subset X$ (example below).

Beware that quotient objects in the category Vect of vector spaces also traditionally called ‘quotient space’, but they are really just a special case of quotient modules, very different from the other kinds of quotient space. However in topological vector spaces both concepts come together.

Definitions

In $Top$

Definition

(quotient topological space)

Let $(X,\tau_X)$ be a topological space and let

$$R_\sim \subset X \times X$$

be an equivalence relation on its underlying set. Then the quotient topological space has

• as underlying set the quotient set $X/\sim$, hence the set of equivalence classes,

and

• a subset $O \subset X/\sim$ is declared to be an open subset precisely if its preimage $\pi^{-1}(O)$ under the canonical projection map

  $$\pi \;\colon\; X \to X/\sim$$

  is open in $X$.

To see that this indeed does define a topology on $X/\sim$ it is sufficient to observe that taking pre-images commutes with taking unions and with taking intersections.

Often one considers this with input datum not the equivalence relation, but any surjection

$$\pi \;\colon\; X \longrightarrow Y$$

of sets. Of course this identifies $Y = X/\sim$ with $(x_1 \sim x_2) \Leftrightarrow (\pi(x_1) = \pi(x_2))$. Hence the quotient topology on the codomain set of a function out of any topological space has as open subsets those whose pre-images are open.

Equivalently this is the final topology or strong topology induced on $Y$ by the function $X \to Y$, see at Top – Universal constructions.

For this construction the function $X \to Y$ need not even be surjective, and we could generalize to a sink instead of a single map; in such a case one generally says final topology or strong topology. See also at topological concrete category.

In $Loc$

A quotient space in $Loc$ is given by a regular subobject in Frm.

(More details needed.)

Examples

Example

The trigonometric function

$$(cos(-),sin(-)) \;\colon\; [0,2\pi] \longrightarrow S^1 \subset \mathbb{R}^2$$

from the closed interval with its Euclidean metric topology to the unit circle equipped with the subspace topology of the Euclidean plane

descends to a homeomorphism on the quotient space $[0,2 \pi]/(0 \sim 2 \pi)$ by the equivalence relation which identifies the two endpoints of the open interval.

$$\array{ [0,2\pi] &\overset{(cos(-),sin(-))}{\longrightarrow}& S^1 \\ \downarrow & \nearrow_{\mathrlap{\simeq}} \\ [0,2\pi]/\simeq } \,.$$

Proof

By the universal property of the quotient it follows that $[0,2\pi]/(0 \sim 2\pi) \to S^1$ is a continuous function. Moreover, it is a bijection on the underlying sets by the $2\pi$-periodicity of sine and coside. Hence it is sufficient to see that it is an open map (by this prop.).

Since the open subsets of $[0,2\pi]$ are unions of

1. the open intervals $(a,b)$ with $0 \lt a \lt b \lt 2\pi$,

2. the half-open intervals $[0,b)$ and $(a,2\pi]$ with $0 \lt a,b \lt 2\pi$

and since the projection map $\pi \colon [0,2\pi] \to [0,2\pi]/(0 \sim 2\pi)$ is injective on $(0, 2\pi)$, the open subsets of $[0,2\pi]/(0 \sim 2\pi)$ are unions of

1. the open intervals $(a,b)$ with $0 \lt a \lt b \lt 2\pi$,

2. the glued half-open intervals $(b,2\pi]/(0\sim 2\pi) \cup [0,a)/(0 \sim 2\pi)$ for $0 \lt a,b \lt 2\pi$.

By the $2\pi$-periodicity of $(cos(-),sin(-))$, the image of the latter under $(cos(-),sin(-))$ is the same as the image of $(b, 2\pi + a)$. Since the function $(cos(-),sin(-)) \colon \mathbb{R} \to S^1$ is clearly an open map, it follows that the images of these open subsets in $S^1$ are open.

Example

(quotient by a subspace)

Let $X$ be a topological space and $A \subset X$ a non-empty subset. Consider the equivalence relation on $X$ which identifies all points in $A$ with each other. The resulting quotient space (def. ) is often simply denoted $X/A$.

Notice that $X/A$ is canonically a pointed topological space, with base point the equivalence class $A/A \subset X/A$ of $A$.

If $A = \emptyset$ is the empty space, then one defines

$$X/\emptyset \coloneqq X_+ \coloneqq X \sqcup \ast$$

to be the disjoint union space of $X$ with the point space. This is no longer a quotient space, but both constructions are unified by the pushout $i \colon A \to X$ along the map $A \to \ast$, equivalently the cokernel of the inclusion:

$$\array{ A &\overset{i}{\hookrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& X/A } \,.$$

This kind of quotient space plays a central role in the discussion of long exact sequences in cohomology, see at generalized (Eilenberg-Steenrod) cohomology.

Example

Consider the real numbers $\mathbb{R}$ equipped with their Euclidean metric topology. Consider on $\mathbb{R}$ the equivalence relation which identifies all real numbers that differ by a rational number:

$$(x_1 \sim_{\mathbb{Q}} x_2) \Leftrightarrow \left( x_2 - x_1 \in \mathbb{Q} \subset \mathbb{Q} \right) \,.$$

Then the quotient space $\mathbb{R}/\sim_{\mathbb{Q}}$ is a codiscrete topological space.

Proof

We need to check that the only open subsets of $X/\sim_{\mathbb{Q}}$ are the empty set and the entire set $X/\sim_{\mathbb{Q}}$.

So let $U \subset \mathbb{R}/\sim$ be a non-empty subset.

Write $\pi \colon \mathbb{R} \to \mathbb{R}/\sim_{\mathbb{Q}}$ for the quotient projection. By definition $U$ is open precisely if its pre-image $\pi^{-1}(U) \subset \mathbb{R}$ is open. By the Euclidean topology, this is the case precissely if $\pi^{-1}(U)$ is a union of open intervals. Since by assumption $\pi^{-1}(U)$ is non-empty, it contains at least one open interval $(a,b) \subset \mathbb{R}$, with $a \lt b$. By the density of the rational numbers, there exists a rational number $q \in \mathbb{Q} \subset \mathbb{R}$ with

$$0 \lt q \lt b - a \,.$$

By definition of $\sim_{\mathbb{Q}}$ we have for all $n \in \mathbb{Z}$ that all elements in $(a + n q, b + n q) \subset \mathbb{R}$ are $\sim_{\mathbb{Q}}$-equivalent to elements in $(a,b)$, hence that also $(a+q,b+q) \subset \pi^{-1}(U)$. But the union of these open intervals is all of $\mathbb{R}$

$$\underset{n \in \mathbb{Z}}{\cup} (q + n q, b + n q) \;=\; \mathbb{R}$$

and so $\pi^{-1}(U) = \mathbb{R}$.

Properties

1. Recall that a map $q \colon X \to Y$ is open if $q(U)$ is open in $Y$ whenever $U$ is open in $X$. It is not the case that a quotient map $q \colon X \to Y$ is necessarily open. Indeed, the identification map $q \colon I \sqcup \{\ast\} \to S^1$, where the endpoints of $I$ are identified with $\ast$, takes the open point $\ast$ of the domain to a non-open point in $S^1$.

2. Nor is it the case that a quotient map is necessarily a closed map; the classic example is the projection map $\pi_1 \colon \mathbb{R}^2 \to \mathbb{R}$, which projects the closed locus $x y = 1$ onto a non-closed subset of $\mathbb{R}$. (This is a quotient map, by the next remark.)

3. It is easy to prove that a continuous open surjection $p \colon X \to Y$ is a quotient map. For instance, projection maps $\pi \colon X \times Y \to Y$ are quotient maps, provided that $X$ is inhabited. Likewise, a continuous closed surjection $p: X \to Y$ is a quotient map: $p^{-1}(U)$ is open $\Rightarrow$ $p^{-1}(\neg U)$ is closed $\Rightarrow$ $p(p^{-1}(\neg U)) = \neg U$ is closed $\Rightarrow$ $U$ is open. For example, a continuous surjection from a compact space to a Hausdorff space is a quotient map.

Proposition

A continuous function

$$f \;\colon\; (X, \tau_X) \longrightarrow (Y,\tau_Y)$$

whose underlying function $f \colon X \longrightarrow Y$ is surjective exhibits $\tau_Y$ as the corresponding quotient topology precisely if $f$ sends open and $f$-saturated subsets in $X$ to open subsets of $Y$. By this lemma this is the case precisely if it sends closed and $f$-saturated subsets to closed subsets.

Related concepts

• Sullivan model of finite G-quotient?

• image

• subspace, subframe

• quotient, quotient stack, quotient type

examples of universal constructions of topological spaces:

$\phantom{AAAA}$limits	$\phantom{AAAA}$colimits
$\,$ point space$\,$	$\,$ empty space $\,$
$\,$ product topological space $\,$	$\,$ disjoint union topological space $\,$
$\,$ topological subspace $\,$	$\,$ quotient topological space $\,$
$\,$ fiber space $\,$	$\,$ space attachment $\,$
$\,$ mapping cocylinder, mapping cocone $\,$	$\,$ mapping cylinder, mapping cone, mapping telescope $\,$
	$\,$ cell complex, CW-complex $\,$