Semitopological group
From Groupprops
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
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 ![]() ![]() ![]() |
Separate continuity is weaker than joint continuity, which would mean continuity from ![]() ![]() ![]() |
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 ![]() |
|
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