Topological group: Difference between revisions

Latest revision as of 17:04, 2 August 2012

This article gives a basic definition in the following area: topological group theory
View other basic definitions in topological group theory |View terms related to topological group theory |View facts related to topological group theory

This article describes a compatible combination of two structures: group and topological space

This article defines the notion of group object in the category of topological spaces|View other types of group objects

This is a variation of group|Find other variations of group | Read a survey article on varying group

Definition

Abstract definition

The notion of topological group can be defined in the following equivalent ways:

In the language of universal algebra, it is a group equipped with a topology for which all the defining operations of groups are continuous.
It is a group object in the category of topological spaces.

Concrete definition

A topological group is a set $G$ endowed with the following two structures:

The structure of a group, i.e., a binary operation called multiplication or product, a constant called the identity element, and a unary operation called the inverse map, and satisfying the conditions for a group
The structure of a topological space

such that the following compatibility conditions are satisfied:

Operation	Arity	Condition	Comments
Multiplication or product	2 (so it's a map $G \times G \to G$ )	continuous as a map from $G \times G$ (equipped with the product topology) to $G$ . In other words, the group multiplication $(g, h) \mapsto g h$ is jointly continuous.	Joint continuity is strictly stronger than separate continuity, which would mean continuity in each input holding the other input fixed.
Identity element	0 (it's a constant element $e \in G$ )	no condition. As such, we may impose the condition that the map from a one-point space to $G$ sending the point to the identity element is continuous, but this condition is vacuously true.
Inverse map	1 (so it's a map $G \to G$ )	continuous as a map from $G$ to itself with the equipped topology. In other words, $g \mapsto g^{- 1}$ is continuous.	Note that because the inverse map is its own inverse (see inverse map is involutive), this is equivalent to it being a self-homeomorphism of $G$ .

T0 assumption

Some people assume a topological group to be $T_{0}$ , that is, that there is no pair of points with each in the closure of the other. This is not a very restrictive assumption, because if we quotient out a topological group by the closure of the identity element, we do get a $T_{0}$ -topological group. However, the definition above does not include this assumption. Further information: T0 topological group

Caution about algebraic groups

Further information: Algebraic groups are not topological groups

Algebraic groups can be given a topology arising from their algebraic variety structure, namely, the Zariski topology. However, with some trivial exceptions, algebraic groups are not topological groups. The reason is that the multiplication map is not jointly continuous. Another way of seeing this is that algebraic groups are $T_{1}$ but not Hausdorff under the Zariski topology, but we know that for topological groups, being $T_{1}$ is equivalent to being Hausdorff.

However, there are two saving graces:

Algebraic groups are quasitopological groups, where a quasitopological group is a group with a topology where the inverse map is continuous and the group multiplication map is separately continuous in each variable.
Algebraic groups over a topological field are topological groups with respect to the topology arising from the field topology, and in the case that the field comes equipped with analytic structure, they become Lie groups over the field. For instance, real algebraic groups are real Lie groups and complex algebraic groups are complex Lie groups.

Relation with other structures

Stronger structures

Lie group

Weaker structures

Structure	Meaning	Intermediate notions
paratopological group	multiplication is still required to be jointly continuous, but there is no condition on the inverse map.
quasitopological group	multiplication is required to only be separately continuous (which is weaker than jointly continuous), and the inverse map is required to be continuous.
semitopological group	multiplication is required to only be separately continuous (which is weaker than jointly continuous), and there is no condition on the inverse map.	\|FULL LIST, MORE INFO
H-group	topological space with binary operation, constant element, and inverse map, all continuous, satisfying the group axioms up to homotopy	\|FULL LIST, MORE INFO

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 145, Supplementary Exercises (assumes $T_{0}$ in the definition of topological group)

@@ Line 1: / Line 1: @@
 {{basicdef in|topological group theory}}
 {{compatiblecombination|group|topological space}}
+{{group object|topological space}}
+{{variation of|group}}
-{{group object|topological space}}
 ==Definition==
-===Symbol-free definition===
+===Abstract definition===
+The notion of topological group can be defined in the following equivalent ways:
+# In the language of universal algebra, it is a group equipped with a topology for which all the defining operations of groups are continuous.
+# It is a [[group object]] in the [[category of topological spaces]].
+===Concrete definition===
-A '''topological group''' is a set endowed with the following two structures:
+A '''topological group''' is a set <math>G</math> endowed with the following two structures:
-* The structure of a [[group]], viz a binary operation called multiplication or product, a unary operation called the inverse map, and a constant called the identity element satisfying the conditions for a group
+* The structure of a [[group]], i.e., a binary operation called multiplication or product, a constant called the identity element, and a unary operation called the inverse map, and  satisfying the conditions for a group
 * The structure of a [[topological space]]
 such that the following compatibility conditions are satisfied:
-* The inverse map is a continuous map from the group to itself (as a topological space map)
+{| class="sortable" border="1"
-* The group multiplication map is a [[jointly continuous map]] i.e. a continuous map from the Cartesian product of the group with itself, to the group (where the Cartesian product is given the product topology).
+! Operation !! Arity !! Condition !! Comments
+|-
+| Multiplication or product || 2 (so it's a map <math>G \times G \to G</math>) || '''continuous''' as a map from <math>G \times G</math> (equipped with the product topology) to <math>G</math>. In other words, the group multiplication <math>(g,h) \mapsto gh</math> is jointly continuous. || Joint continuity is strictly stronger than ''separate'' continuity, which would mean continuity in each input holding the other input fixed.
+|-
+| Identity element || 0 (it's a constant element <math>e \in G</math>) || '''no condition'''. As such, we may impose the condition that the map from a [[one-point space]] to <math>G</math> sending the point to the identity element is continuous, but this condition is vacuously true. ||
+|-
+| Inverse map || 1 (so it's a map <math>G \to G</math>) || '''continuous''' as a map from <math>G</math> to itself with the equipped topology. In other words, <math>g \mapsto g^{-1}</math> is continuous. || Note that because the inverse map is its own inverse (see [[inverse map is involutive]]), this is equivalent to it being a self-homeomorphism of <math>G</math>.
+|}
+===T0 assumption===
-Some people assume a topological group to be <math>T_0</math>, that is, that there is no pair of points with each in the closure of the other. This is not a very restrictive assumption, because if we quotient out a topological group by the closure of the identity element, we do get a <math>T_0</math>-topological group. {{further|[[T0 topological group]]}}
+Some people assume a topological group to be <math>T_0</math>, that is, that there is no pair of points with each in the closure of the other. This is not a very restrictive assumption, because if we quotient out a topological group by the closure of the identity element, we do get a <math>T_0</math>-topological group. However, the definition above does not include this assumption. {{further|[[T0 topological group]]}}
-===Definition with symbols===
+==Caution about algebraic groups==
-A '''topological group''' is a set <math>G</math> endowed with two structures:
+{{further|[[Algebraic groups are not topological groups]]}}
-* The structure of a [[group]] viz a multiplication <math>*</math> and an inverse map <math>g \mapsto g^{-1}</math> and an identtiy element <math>e</math>.
+[[Algebraic group]]s can be given a topology arising from their algebraic variety structure, namely, the Zariski topology. However, with some trivial exceptions, algebraic groups are ''not'' topological groups. The reason is that the multiplication map is not jointly continuous. Another way of seeing this is that algebraic groups are <math>T_1</math> but not Hausdorff under the Zariski topology, but we know that for topological groups, being <math>T_1</math> is equivalent to being Hausdorff.
-* The structure of a [[topological space]] viz a topology <math>\tau</math>
-such that:
+However, there are two saving graces:
-* <math>g \mapsto g^{-1}</math> is a [[continuous map]] with respect to <math>\tau</math>.
+* [[Algebraic groups are quasitopological groups]], where a [[quasitopological group]] is a group with a topology where the inverse map is continuous and the group multiplication map is ''separately'' continuous in each variable.
-* <math>(g,h) \mapsto g * h</math> is a [[jointly continuous map]] viz it is a continuous map from <math>G \times G</math> with the product topology, to <math>G</math>.
+* Algebraic groups over a [[topological field]] are topological groups with respect to the topology arising from the ''field'' topology, and in the case that the field comes equipped with analytic structure, they become [[Lie group]]s over the field. For instance, real algebraic groups are real Lie groups and complex algebraic groups are complex Lie groups.
 ==Relation with other structures==
@@ Line 36: / Line 51: @@
 ===Stronger structures===
-* [[Algebraic group]] over a [[topological field]]
 * [[Lie group]]
 ===Weaker structures===
-* [[Semitopological group]]
+{| class="sortable" border="1"
-* [[Left-topological group]]
+! Structure !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Right-topological group]]
+|-
+| [[Stronger than::paratopological group]] || multiplication is still required to be jointly continuous, but there is no condition on the inverse map. || || ||
+|-
+| [[Stronger than::quasitopological group]] || multiplication is required to only be separately continuous (which is weaker than jointly continuous), and the inverse map is required to be continuous. || || ||
+|-
+| [[Stronger than::semitopological group]] || multiplication is required to only be separately continuous (which is weaker than jointly continuous), and there is no condition on the inverse map. || || || {{intermediate notions short|semitopological group|topological group}}
+|-
+| [[Stronger than::H-group]] || topological space with binary operation, constant element, and inverse map, all continuous, satisfying the group axioms up to homotopy || || || {{intermediate notions short|H-group|topological group}}
+|}
+==References==
+===Textbook references===
+* {{booklink|Munkres}}, Page 145, Supplementary Exercises (assumes <math>T_0</math> in the definition of topological group)