nLab
quotient space

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


Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Geometry

Quotient spaces

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/AX/A by a subspace AXA \subset X (example 2 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 TopTop

Definition

(quotient topological space)

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

R X×X R_\sim \subset X \times X

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

and

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

    π:XX/ \pi \;\colon\; X \to X/\sim

    is open in XX.

To see that this indeed does define a topology on X/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

π:XY \pi \;\colon\; X \longrightarrow Y

of sets. Of course this identifies Y=X/Y = X/\sim with (x 1x 2)(π(x 1)=π(x 2))(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 YY by the function XYX \to Y, see at Top – Universal constructions.

For this construction the function XYX \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 LocLoc

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

(More details needed.)

Examples

Example

The trigonometric function

(cos(),sin()):[0,2π]S 1 2 (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 [0,2π]/(02π)[0,2 \pi]/(0 \sim 2 \pi) by the equivalence relation which identifies the two endpoints of the open interval.

[0,2π] (cos(),sin()) S 1 [0,2π]/. \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 [0,2π]/(02π)S 1[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 2π2\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 [0,2π][0,2\pi] are unions of

  1. the open intervals (a,b)(a,b) with 0<a<b<2π0 \lt a \lt b \lt 2\pi,

  2. the half-open intervals [0,b)[0,b) and (a,2π](a,2\pi] with 0<a,b<2π0 \lt a,b \lt 2\pi

and since the projection map π:[0,2π][0,2π]/(02π)\pi \colon [0,2\pi] \to [0,2\pi]/(0 \sim 2\pi) is injective on (0,2π)(0, 2\pi), the open subsets of [0,2π]/(02π)[0,2\pi]/(0 \sim 2\pi) are unions of

  1. the open intervals (a,b)(a,b) with 0<a<b<2π0 \lt a \lt b \lt 2\pi,

  2. the glued half-open intervals (b,2π]/(02π)[0,a)/(02π)(b,2\pi]/(0\sim 2\pi) \cup [0,a)/(0 \sim 2\pi) for 0<a,b<2π0 \lt a,b \lt 2\pi.

By the 2π2\pi-periodicity of (cos(),sin())(cos(-),sin(-)), the image of the latter under (cos(),sin())(cos(-),sin(-)) is the same as the image of (b,2π+a)(b, 2\pi + a). Since the function (cos(),sin()):S 1(cos(-),sin(-)) \colon \mathbb{R} \to S^1 is clearly an open map, it follows that the images of these open subsets in S 1S^1 are open.

Example

(quotient by a subspace)

Let XX be a topological space and AXA \subset X a non-empty subset. Consider the equivalence relation on XX which identifies all points in AA with each other. The resulting quotient space (def. 1) is often simply denoted X/AX/A.

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

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

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

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

A i X (po) * X/A. \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 x 2)(x 2x 1). (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/ X/\sim_{\mathbb{Q}} are the empty set and the entire set X/ X/\sim_{\mathbb{Q}}.

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

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

0<q<ba. 0 \lt q \lt b - a \,.

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

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

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

Properties

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

  2. Nor is it the case that a quotient map is necessarily a closed map; the classic example is the projection map π 1: 2\pi_1 \colon \mathbb{R}^2 \to \mathbb{R}, which projects the closed locus xy=1x 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:XYp \colon X \to Y is a quotient map. For instance, projection maps π:X×YY\pi \colon X \times Y \to Y are quotient maps, provided that XX is inhabited. Likewise, a continuous closed surjection p:XYp: X \to Y is a quotient map: p 1(U)p^{-1}(U) is open \Rightarrow p 1(¬U)p^{-1}(\neg U) is closed \Rightarrow p(p 1(¬U))=¬Up(p^{-1}(\neg U)) = \neg U is closed \Rightarrow UU is open. For example, a continuous surjection from a compact space to a Hausdorff space is a quotient map.

Proposition

A continuous function

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

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

examples of universal constructions of topological spaces:

AAAA\phantom{AAAA}limitsAAAA\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 \,

Revised on June 14, 2017 05:15:10 by Urs Schreiber (131.220.184.222)