mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a semitopological group is a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

with a

group action In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...

that is continuous with respect to each variable considered separately. It is a weakening of the concept of a

topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...

; 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^

is also required to be continuous. Here

G \times G

is viewed as a topological space with the

product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...

.) Clearly, every topological group is a semitopological group. To see that the converse does not hold, consider the

real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...

(\mathbb,+)

with its usual structure as an additive

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

. Apply the

lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on \mathbb, the set of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of in ...

\mathbb

with topological basis the family

\

. Then

g_1

is continuous, but

g_2

is not continuous at 0:

,b)

is an open neighbourhood of 0 but there is no neighbourhood of 0 contained in

g_2^([0,b))

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

References

{{Reflist Topological groups

Formal definition

See also

References