Paratopological group
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This article gives a basic definition in the following area: topological group theory
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This article describes a compatible combination of two structures: group and topological space
Definition
A paratopological group is a set endowed with the following two structures:
- The structure of a group, viz., an associative binary operation with identity element and inverses
- The structure of a topological space
such that the following are true:
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 ) | no condition. In particular, we do not require this operation to be continuous. | If we did impose the additional condition that the inverse map be continuous, we would get the definition of topological group. |
Relation with other structures
Stronger structures
Structure | Meaning | Proof of implication | Proof of strictness (reverse implication failure) |
---|---|---|---|
topological group | group multiplication is jointly continuous and inverse map is continuous | (obvious) | paratopological group not implies topological group |
Weaker structures
Structure | Meaning |
---|---|
semitopological group | group multiplication is separately continuous in both variables |
left-topological group | group multiplication is continuous in right input, i.e., left multiplication maps are continuous |
right-topological group | group multiplication is continuous in left input, i.e., right multiplication maps are continuous |
Incomparable structures
- Quasitopological group means the group multiplication is separately continuous but the inverse map is required to be continuous.