In mathematics, a semitopological group is a topological space with a group operation 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 {\displaystyle G} is a topological space that is also a group such that
g 1 : G × G → G : ( x , y ) ↦ x y {\displaystyle g_{1}:G\times G\to G:(x,y)\mapsto xy}
is continuous with respect to both x {\displaystyle x} and y {\displaystyle y} . (Note that a topological group is continuous with reference to both variables simultaneously, and g 2 : G → G : x ↦ x − 1 {\displaystyle g_{2}:G\to G:x\mapsto x^{-1}} is also required to be continuous. Here G × G {\displaystyle 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 ( R , + ) {\displaystyle (\mathbb {R} ,+)} with its usual structure as an additive abelian group. Apply the lower limit topology to R {\displaystyle \mathbb {R} } with topological basis the family { [ a , b ) : − ∞ < a < b < ∞ } {\displaystyle \{[a,b):-\infty <a<b<\infty \}} . Then g 1 {\displaystyle g_{1}} is continuous, but g 2 {\displaystyle g_{2}} is not continuous at 0: [ 0 , b ) {\displaystyle [0,b)} is an open neighbourhood of 0 but there is no neighbourhood of 0 contained in g 2 − 1 ( [ 0 , b ) ) {\displaystyle 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.
See also Lie group Algebraic group Compact group Topological ring
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