Paratopological group not implies topological group

Statement

There are two versions:

• Weak version: It is possible to construct a paratopological group that is not a topological group.
• Strong version: It is possible to construct a topological space that occurs as the underlying space of a paratopological group but does not occur as the underlying space of any topological group.

Proof

Example of the Sorgenfrey line

Further information: Sorgenfrey line

References

• Does the Sorgenfrey Line have a group operation compatible with its order topology?