nLab
quotient space

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

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.

Quotient objects in the category VectVect of vector spaces also traditionally use the term ‘quotient space’, but they are really just a special case of quotient modules, very different from the other kinds of quotient space. Quotient TVSes, however, combine both aspects.

Definitions

In TopTop

Let XX be a topological space and \sim an equivalence relation on (the underlying set of) XX. (Since monomorphisms in Top are just injective continuous maps, to give an equivalence relation on the underlying set of a topological space is the same as to give a congruence on that space in TopTop.) Let Y=X/Y = X/{\sim} be the quotient set and q:XYq\colon X\to Y the quotient map.

The quotient topology, or identification topology, induced on YY from XX says that a subset UYU\subseteq Y is open if and only if the preimage q 1(U)Xq^{-1}(U)\subseteq X is open. With this topology YY is a quotient space or identification space of XX.

Obviously, up to homeomorphism, all that matters is the surjective function XYX\to Y. For the above definition, we don’t even need it to 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 topological concrete category.

In LocLoc

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

(More details needed.)

Properties

Definition

(saturated subset)

Let f:XYf \;\colon\; X \longrightarrow Y be a function of sets. Then a subset SSS \subset S is called an ff-saturated subset (or just separated subset, if ff is understood) if SS is the pre-image of its image:

(SXf-saturated)(S=f 1(f(S))). \left(S \subset X \,\, f\text{-saturated} \right) \,\Leftrightarrow\, \left( S = f^{-1}(f(S)) \right) \,.

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.

Revised on April 28, 2017 06:28:05 by Urs Schreiber (92.218.150.85)