Topology quotient

Introduction

Many interesting spaces, the circle, the tori, the ribbon of Möbius, the projective spaces are defined like quotients. Topology quotient often provides the way most natural to provide a definite unit " géométriquement" of a natural topology. Let us quote for example (see low) the whole of the vectorial subspaces of dimension $p\; \backslash ,$ of $\backslash \; mathbb\; \{R\}\; ^n\; \backslash ,$.

Let us quote also the case of surfaces of $\backslash \; mathbb\; \{R\}\; ^3$ particular: tori with $p\; \backslash ,$ holes. To formalize this concept, it is necessary to define the operation consisting in to add a handle to a surface. That is done without too much difficulty by using topology quotient, whereas it is not obvious whole to define such surfaces by an equation! This concept illustrates also the effectiveness of general topology compared to the theory of the metric Espaces, often used like intoduction with Topology: well that the topology of the majority of the examples described below can be defined by metric, such metric is not always easy with constuire.

Definition and principal properties

That is to say $X\; \backslash ,$ a topological Space and $\backslash \; mathcal\; \{R\}\; \backslash ,$ a Relation of equivalence on $X\; \backslash ,$. One will note $p\; \backslash ,$ the natural application of $X\; \backslash ,$ in $X/\backslash \; mathcal\; \{R\}\; \backslash ,$ which associates with an element $X\; \backslash ,$ its class of equivalence.

Topology quotient on $X/\backslash \; mathcal\; \{R\}\; \backslash ,$ is in the following way defined: so that a part $U\; \backslash \; subset\; X\; \backslash \; mathcal\; \{R\}\; \backslash ,$ is opened, it is necessary and is enough that $p^\; \{-\; 1\}\; (U)\; \backslash ,$ is open in $X\; \backslash ,$. Like, according to the elementary set theory, the reciprocal image of one intersection (resp. of a meeting) is equal to the intersection (resp. the meeting) of reciprocal images, a topology well thus is defined.

That is to say $Y\; \backslash ,$ a topological space unspecified . Then, so that an application $F\; \backslash ,$ of $X/\backslash \; mathcal\; \{R\}\; \backslash ,$ in $Y\; \backslash ,$ is continuous, it is necessary and it is enough that the application $F\; \backslash \; circ\; p\; \backslash ,$ of $X\; \backslash ,$ in $Y\; \backslash ,$ is continuous.

The definition of topology quotient is made precisely so that this property is satisfied: if $V\; \backslash ,$ is open of $Y,$, then $f^\; \{-\; 1\}\; (V)\; \backslash ,$ is open in $X/\backslash \; mathcal\; \{R\}\; \backslash ,$ if and only if $p^\; \{-\; 1\}\; \backslash \; big\; (f^\; \{-\; 1\}\; (V)\; \backslash \; big)\; \backslash ,$ is open in $X\; \backslash ,$. But $p^\; \{-\; 1\}\; \backslash \; big\; (f^\; \{-\; 1\}\; (V)\; \backslash \; big)\; =\; (F\; \backslash \; circ\; p)\; ^\; \{-\; 1\}\; (V)\; \backslash ,$.

Notice

This criterion also says to us that if an application continues $g\; \backslash ,$ of $X\; \backslash ,$ in $Y\; \backslash ,$ is constant on the classes of equivalence, then the application $\backslash \; overline\; G\; \backslash ,$ of $X/\backslash \; mathcal\; \{R\}\; \backslash ,$ in $Y\; \backslash ,$ defined by passage in the quotient is automatically continuous.

Some traps

The price to be paid for the simpicity of this definition is the fact that even if $X\; \backslash ,$ are separated, $X/\backslash \; mathcal\; \{R\}\; \backslash ,$ provided with topology quotient will not be it inevitably (and even if it is it, it will have to be shown individually). Indeed, if $U\; \backslash ,$ is open in $X\; \backslash ,$, there is no reason in general so that $p\; (U)\; \backslash ,$ is open in $X/\backslash \; mathcal\; \{R\}\; \backslash ,$, and if $U\_1\; \backslash ,$ and $U\_2\; \backslash ,$ are two disjoined parts of $X\_,$, their images by $p\; \backslash ,$ are not it necessarily.

First examples

• If $X=\; \backslash ,$, and if $\backslash \; mathcal\; \{R\}$ is the relation of equivalence which

identify $0\; \backslash ,$ and $1\; \backslash ,$, $X/\backslash \; mathcal\; \{R\}\; \backslash ,$ provided with topology quotient is homeomorphic with the circle.

• If $A\; \backslash ,$ is part of $X\; \backslash ,$,

let us note $X/A\; \backslash ,$ the space obtained by identifying all the points of $A\; \backslash ,$, provided with topology quotient.

* If $X\; \backslash ,$ is a closed Euclidean ball of dimension $n\; \backslash ,$ and $A\; \backslash ,$ its border

(which is the sphere unit $S^\; \{n-1\}\; \backslash ,$) one can show that $X/A\; \backslash ,$ is homémorphe with $S^\; \{N\}\; \backslash ,$.

* If $X=\; \backslash \; mathbb\; \{R\}\; \backslash ,$ and $A=\; \backslash \; mathbb\; \{Q\}\; \backslash ,$, topology quotient on $X/A\; \backslash ,$ is coarse topology.

Stickings together

Are $X\; \backslash ,$ and $Y\; \backslash ,$ two spaces topological, $A\; \backslash ,$ part of $X\; \backslash ,$, $B\; \backslash ,$ part of $Y\; \backslash ,$, and $f:\; With\; \backslash \; mapsto\; B\; \backslash ,$ a homeomorphism.

the sticking together of $X\; \backslash ,$ and $Y\; \backslash ,$ along $f\; \backslash ,$ of the disjoined meeting $X\; \backslash \; coprod\; Y\; \backslash ,$ by the relation is the quotient of équvalence which identifies the elements of $A\; \backslash ,$ and those of $B\; \backslash ,$ by means of $f\; \backslash ,$.

One can describe the operation thus consisting in adding a handle to a surface $X\; \backslash ,$. One takes $Y=S^1\; \backslash \; times\; \backslash ,$, $A=\; \backslash \; partial\; D\_1\; \backslash \; cup\; A=\; \backslash \; partial\; D\_2$ for two disjoined closed discs $D\_1$ and $D\_2$, $B=S^1\; \backslash \; times\; \backslash \; \{0\; \backslash \}\; \backslash \; cup\; =S^1\; \backslash \; times\; \backslash \; \{1\; \backslash \}$; $f\; \backslash ,$ of $\backslash \; partial\; D\_1$ on $S^1\; \backslash \; times\; \backslash \; \{0\; \backslash \}$ is a homeomorphism and of $\backslash \; partial\; D\_1$ on $S^1\; \backslash \; times\; \backslash \; \{1\; \backslash \}$. (it is faster to draw than to describe).

Actions of groups

The circle can be also obtained like quotient of $\backslash \; mathbb\; \{R\}\; \backslash ,$ by the relation $\backslash \; mathcal\; \{R\}\; \backslash ,$ defined by $x\; \backslash \; mathcal\; \{R\}\; there\; \backslash \; Longleftrightarrow\; X\; there\; \backslash \; in\; \backslash \; mathbb\; \{Z\}\; \backslash ,$ More generally, it is said that a topological Groupe $\backslash \; Gamma\; \backslash ,$ acts continuously on a topological space $X\; \backslash ,$ if there is a continuous application $(\backslash \; gamma,\; X)\; \backslash \; mapsto\; \backslash \; gamma\; \backslash \; cdot\; x$ of $\backslash \; Gamma\; \backslash \; times\; X\; \backslash ,$ in $X\; \backslash ,$ such as

$\backslash \; gamma^\; \backslash \; premium\; \backslash \; cdot\; (\backslash \; gamma\; \backslash \; cdot\; X)\; =\; (\backslash \; gamma^\; \backslash \; premium\; \backslash \; gamma)\; \backslash \; cdot\; x$ and $e\; \backslash \; cdot\; x=x$

Space quotient by the relation of equivalence

$x\; \backslash \; mathcal\; \{R\}\; there\; \backslash \; Longleftrightarrow\; \backslash \; exists\; \backslash \; gamma\; \backslash \; in\; \backslash \; Gamma,\; x=\; \backslash \; gamma\; \backslash \; cdot\; there$

is noted $X/\backslash \; Gamma\; \backslash ,$, and is called space of the orbits of $\backslash \; Gamma\; \backslash ,$

To avoid too pathological situations, one often supposes that $X\; \backslash ,$ is Localement compact and that the action of $\backslash \; Gamma\; \backslash ,$ clean is , i.e. the reciprocal image of very compact $K\; \backslash \; subset\; X\; \backslash ,$ by the application $(\backslash \; gamma,\; X)\; \backslash \; mapsto\; \backslash \; gamma\; \backslash \; cdot\; x$ is compact. If $\backslash \; Gamma\; \backslash ,$ is a discrete group (frequent and already interesting situation), in other words, the whole of the $\backslash \; gamma\; \backslash ,$ such as $\backslash \; gamma\; (K)\; \backslash \; course\; K\; \backslash \; not=\; \backslash \; emptyset$ is finished.

It is shown that the quotient of a compact space Localement by a clean action is separate (and locally compact).

Examples

• For the action of $\backslash \; mathbb\; \{Z\}$ on $\backslash \; mathbb\; \{R\}\; \backslash \; times\; \backslash ,$

given by $n\; \backslash \; cdot\; (X,\; there)\; =\; (x+n,\; there)$ space quotient is a cylinder. For the action given by $n\; \backslash \; cdot\; (X,\; there)\; =\; (x+n,\; (-\; 1)\; ^ny)$, space quotient is a Ruban of Möbius.

• For the action of $\backslash \; mathbb\; \{Z\}\; ^2\; \backslash ,$ on $\backslash \; mathbb\; \{R\}\; ^2\; \backslash ,$

given by $(m,\; N)\; \backslash \; cdot\; (X,\; there)\; =\; (x+m,\; y+n)$ space quotient is one Torus

• For the action of the group with two elements $\backslash \; \{I,\; \backslash \; sigma\; \backslash \}\; \backslash ,$

on the sphere $S^n\; \backslash ,$ defined by $\backslash \; sigma.x=\; -\; X\; \backslash ,$, the quotient projective Espace is the .

• On $\backslash \; mathbb\; \{R\}\; ^2\; \backslash ,$, the transformations

$(X,\; there)\; \backslash \; mapsto\; (X,\; y+1)\; \backslash ,$ and $(X,\; there)\; \backslash \; mapsto\; (x+1,\; there)\; \backslash ,$ generate a group $\backslash \; Gamma\; \backslash ,$ which acts (it properly is a sub-group discrete group of the isométries euclidennes). The quotient $\backslash \; mathbb\; \{R\}\; ^2/\backslash \; Gamma\; \backslash ,$ is the Bouteille of Klein.

Homogeneous spaces

One thus calls a unit provided with a transitive action of a group $G\; \backslash ,$.

General information

That is to say $G\; \backslash ,$ a topological Group and $G\; \backslash ,$ a sub-group (by inevitably normal). The whole of the classes on the right of $G\; \backslash ,$ modulo $H\; \backslash ,$, noted $G/H\; \backslash ,$, is the quotient of $G\; \backslash ,$ by the relation of equivalence $x^\; \{-\; 1\}\; there\; \backslash \; in\; H\; \backslash ,$. It is also the whole of the orbits of the action of $H\; \backslash ,$ on $G\; \backslash ,$ by translations on right-hand sides.

Proposal. If $H\; \backslash ,$ is closed in $G\; \backslash ,$, $G/H\; \backslash ,$ is separate.

Proof. Like higher, let us indicate by $p:\; G\; \backslash \; rightarrow\; G/H\; \backslash ,$ the application of passage to the quotient. Are $a\; \backslash ,$ and $b\; \backslash ,$ in $G\; \backslash ,$ such as $p\; (a)\; \backslash \; not=p\; (b)\; \backslash ,$, in other words such as $a^\; \{-\; 1\}\; B\; \backslash \; notin\; H\; \backslash ,$. As by assumption $G\; \backslash \; setminus\; H\; \backslash ,$ is opened, it exists, because of continuity of $(X,\; there)\; \backslash \; mapsto\; x^\; \{-\; 1\}\; there\; \backslash ,$ of the open ones $U\; \backslash ,$ and $V\; \backslash ,$, containing $a\; \backslash ,$ respectively and $b\; \backslash ,$, such as, whatever $x\; \backslash \; in\; U\; \backslash ,$ and $y\; \backslash \; in\; U\; \backslash ,$, $x^\; \{-\; 1\}\; there\; \backslash \; notin\; H\; \backslash ,$. Then $U\; \backslash ,$ and $V\; \backslash ,$ do not contain elements equivalents, therefore $p\; (U)\; \backslash ,$ and $p\; (V)\; \backslash ,$ are disjoined (and contain respectively $p\; (a)\; \backslash ,$ and $p\; (b)\; \backslash ,$). Moreover, they are the open ones in $G/H\; \backslash ,$. Indeed, according to the definition of topology quotient, it is enough to check that $p^\; \{-\; 1\}\; \backslash \; big\; (p\; (U)\; \backslash \; big)\; \backslash ,$ and $p^\; \{-\; 1\}\; \backslash \; big\; (p\; (V)\; \backslash \; big)\; \backslash ,$ is it. But $p^\; \{-\; 1\}\; \backslash \; big\; (p\; (U)\; \backslash \; big)\; =U\; \backslash \; cdot\; H\; \backslash ,$ is open like meeting of open.

In premium, so moreover $G\; \backslash ,$ is (locally) compact, it east is the same of $G/H\; \backslash ,$.

Examples

They all are founded on the same principle. That is to say $X\; \backslash ,$ a topological space on which a group (topoldgic) $G\; \backslash ,$ acts transitively. If $a\; \backslash ,$ is a point of $X\; \backslash ,$ given once and for all, the sub-group $U=\; \backslash \; \{G\; \backslash \; in\; G,\; G\; \backslash \; cdot\; a=a\; \backslash ,$ is closed, as soon as $X\; \backslash ,$ is separate. There is a bijection enters $X\; \backslash ,$ and $G/H\; \backslash ,$. One can thus transport with $X\; \backslash ,$ topology quotient of $G/H\; \backslash ,$. (There is a bijection continues $X\; \backslash ,$ - provided with the starting topology - on $G/H\; \backslash ,$, which is a homeomorphism if $X\; \backslash ,$ is compact).

That is to say $\backslash \; mathbb\; \{R\}\; ^n\; \backslash ,$ provided with its usual struture euclidenne and $G=O\; (N)\; \backslash ,$ the orthogonal Group. What precedes applies to the following situations:

• the whole of the orthonormés systems of $k\; \backslash ,$

vectors of $\backslash \; mathbb\; \{R\}\; ^n\; \backslash ,$ is identified with $O\; (N)\; /O\; (n-k)\; \backslash ,$. It is a compact space, and even a variety (called variety of Stiefel).

• the whole of the vectorial subspaces of dimension $k\; \backslash ,$

be identified with $O\; (N)\; /O\; (K)\; \backslash \; times\; O\; (n-k)\; \backslash ,$ It is also a compact space and a variety, called grasmannienne.

Similar geometrical considerations make it possible to see the line whole closely connected of $\backslash \; mathbb\; \{R\}\; ^n\; \backslash ,$ like a homogeneous space.

See too

R. Mneimné, F. Testard, Introduction to the theory of the traditional groups of Dregs, Hermann 1986, ISBN 2 7056 6040 2

J. Lafontaine, Introduction to the differential varieties (CH. 4), ISBN 2 7061 0654 9

Internal bonds

• topological Space
• topological groups
• groups of Dregs

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