Semitopological group
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This article gives a basic definition in the following area: topological group theory
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Definition
A semitopological 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
satisfying the following compatibility conditions:
Operation | Arity | Condition | Comments |
---|---|---|---|
Multiplication or product | 2 (so it's a map ) | separately continuous as a map from to . In other words, the group multiplication is continuous in each variable holding the other variable fixed. | Separate continuity is weaker than joint continuity, which would mean continuity from equipped with the product topology to . Note that the separately continuity condition is equivalent to the left multiplication map and right multiplication map by any element being self-homeomorphisms of . |
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 the operation to be continuous. |
Relation with other structures
Stronger structures
Structure | Meaning |
---|---|
topological group | the multiplication is jointly continuous and the inverse map is continuous |
paratopological group | the multiplication is jointly continuous |
quasitopological group | the multiplication is separately continuous and the inverse map is continuous |
algebraic group |
Weaker structures
Structure | Meaning |
---|---|
left-topological group | the multiplication map is continuous in its right input. Equivalently, the left multiplication map by any element is continuous. |
right-topological group | the multiplication map is continuous in its left input. Equivalently, the right multiplication map by any element is continuous. |
Facts
- Algebraic groups are semitopological groups (in fact, they are quasitopological groups) but they are not necessarily topological groups.
- Locally compact Hausdorff semitopological group implies topological group