Semitopological group
# Semitopological group

Algebraic structure → Group theory Group theory :

Basic notions

Subgroup

Normal subgroup

Quotient group

(Semi-)direct product

Group homomorphisms

kernel

image

direct sum

wreath product

simple

finite

infinite

continuous

multiplicative

additive

cyclic

abelian

dihedral

nilpotent

solvable

action

Glossary of group theory

List of group theory topics :

Finite groups

Classification of finite simple groups

cyclic

alternating

Lie type

sporadic

Cauchy's theorem

Lagrange's theorem

Sylow theorems

Hall's theorem

p-group

Elementary abelian group

Frobenius group

Schur multiplier

Symmetric group $S_n$

Klein four-group V

Dihedral group $D_n$

Quaternion group Q

Dicyclic group $Dic_n$ :

Discrete groupsLattices

Integers ($\mathbb {Z}$)

Free group

Modular groups PSL(2, $\mathbb {Z}$)SL(2, $\mathbb {Z}$)

Arithmetic group

Lattice

Hyperbolic group :

Topological and Lie groups

Solenoid

Circle

General linear GL(n)

Special linear SL(n)

Orthogonal O(n)

Euclidean E(n)

Special orthogonal SO(n)

Unitary U(n)

Special unitary SU(n)

Symplectic Sp(n)

$G_2$

$F_4$

$E_6$

$E_7$

$E_8$

Lorentz

Poincaré

Conformal

Diffeomorphism

Loop

Infinite dimensional Lie group O(∞)SU(∞)Sp(∞) :

Algebraic groups

Linear algebraic group

Reductive group

Abelian variety

Elliptic curve :

vte :

In mathematics, a semitopological group is a topological space with a group action that is continuous with respect to each variable considered separately. It is a weakening of the concept of a topological group; all topological groups are semitopological groups but the converse does not hold.

## Formal definition

A semitopological group $G$ is a topological space that is also a group such that

    $g_{1}:G\times G\to G:(x,y)\mapsto xy$

is continuous with respect to both $x$ and $y$. (Note that a topological group is continuous with reference to both variables simultaneously, and $g_{2}:G\to G:x\mapsto x^{-1}$ is also required to be continuous. Here $G\times G$ is viewed as a topological space with the product topology.)

Clearly, every topological group is a semitopological group. To see that the converse does not hold, consider the real line $(\mathbb {R} ,+)$ with its usual structure as an additive abelian group. Apply the lower limit topology to $\mathbb {R}$ with topological basis the family $\{[a,b):-\infty <a<b<\infty \}$. Then $g_{1}$ is continuous, but $g_{2}$ is not continuous at 0: $[0,b)$ is an open neighbourhood of 0 but there is no neighbourhood of 0 continued in $g_{2}^{-1}([0,b))$.

It is known that any locally compact Hausdorff semitopological group is a topological group. Other similar results are also known.

