Universal covering group of SL(2,R)

Definition
This group is defined in the following equivalent ways:


 * 1) It is the universal covering group of special linear group:SL(2,R). The fundamental group of $$SL(2,\R)$$ is isomorphic to the group of integers, so this group has a central subgroup isomorphic to the group of integers with quotient group $$SL(2,\R)$$.
 * 2) It is the universal covering group of projective special linear group:PSL(2,R). The fundamental group of $$PSL(2,\R)$$ is isomorphic to the group of integers, so this group has a central subgroup isomorphic to the group of integers with quotient group $$PSL(2,\R)$$. Note that since $$PSL(2,\R)$$ is centerless, the center of the universal covering group is precisely this central subgroup.

Structures
The group is a topological group and a real Lie group. It is not a matrix Lie group and it does not have an algebraic group structure (?).