Semitopological group

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 $G\times G\to G$)	separately continuous as a map from $G\times G$ to $G$. In other words, the group multiplication $(g,h)\mapsto gh$ is continuous in each variable holding the other variable fixed.	Separate continuity is weaker than joint continuity, which would mean continuity from $G\times G$ equipped with the product topology to $G$. Note that the separately continuity condition is equivalent to the left multiplication map and right multiplication map by any element being self-homeomorphisms of $G$.
Identity element	0 (it's a constant element $e\in G$)	no condition. As such, we may impose the condition that the map from a one-point space to $G$ sending the point to the identity element is continuous, but this condition is vacuously true.
Inverse map	1 (so it's a map $G\to G$)	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