Topological group
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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 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 ) | continuous as a map from (equipped with the product topology) to . In other words, the group multiplication 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 ) | No condition. As such, we may impose the condition that the map from a one-point space to sending the point to the identity element is continuous, but this condition is vacuously true. | |
Inverse map | 1 (so it's a map ) | continuous as a map from to itself with the equipped topology. In other words, 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 . |
T0 assumption
Some people assume a topological group to be , 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 -topological group. However, the definition above does not include this assumption. Further information: T0 topological group
Definition with symbols
Relation with other structures
Stronger structures
Weaker structures
Structure | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | 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. | Paratopological group, Quasitopological group|FULL LIST, MORE INFO |
References
Textbook references
- Topology (2nd edition) by James R. Munkres^{More info}, Page 145, Supplementary Exercises (assumes in the definition of topological group)