Quasitopological group

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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:

1. The group multiplication is separately continuous in both variables and the inverse map is continuous.
2. 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.
3. 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.
4. 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.
5. 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.
6. For every element of the group, the left multiplication map by that element is continuous, and the inverse map is continuous.
7. 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

Incomparable structures

• Paratopological group: Here, the group multiplication is required to be jointly continuous, but we make no assumption about the continuity of the inverse map.