Quasitopological group
This article gives a basic definition in the following area: topological group theory
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Definition
A quasitopological 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 equivalent conditions:
- The group multiplication is separately continuous in both variables and the inverse map is continuous.
- For every element of the group, the left multiplication map and right multiplication map by that element are continuous, and the inverse map is continuous.
- For every element of the group, the left multiplication map and right multiplication map by that element are self-homeomorphisms, and the inverse map is continuous.
- For every element of the group, the left multiplication map and right multiplication map by that element are self-homeomorphisms, and the inverse map is a self-homeomorphism.
- For every element of the group, the left multiplication map and right multiplication map by that element are continuous, and the inverse map is a self-homeomorphism.
- For every element of the group, the left multiplication map by that element is continuous, and the inverse map is continuous.
- For every element of the group, the right multiplication map by that element is continuous, and the inverse map is continuous.
Relation with other structures
Stronger structures
| Structure | Meaning |
|---|---|
| topological group | multiplication map is jointly continuous (i.e., continuous from the product topology) and inverse map is continuous |
| algebraic group | see algebraic groups are quasitopological groups |
Weaker structures
| Structure | Meaning |
|---|---|
| semitopological group | multiplication map is separately continuous, no assumption about continuity of inverse map |
| left-topological group | multiplication map is continuous in its right input, i.e., left multiplication maps are continuous |
| right-topological group | multiplication map is continuous in its left input, i.e., right multiplication maps are continuous |