We consider the category
Γ
\Gamma
of generalized Lie groups. A generalized Lie group is a topological group
G
G
such that the set
L
G
=
H
o
m
(
R
,
G
)
LG = Hom({\mathbf {R}},G)
of continuous homomorphisms from the reals
R
{\mathbf {R}}
into
G
G
has certain Lie algebra and locally convex topological vector space structures. The full subcategory
Γ
r
{\Gamma ^r}
of
r
r
-bounded (
r
r
positive real number) generalized Lie groups is shown to be left complete. The class of locally compact groups is contained in
Γ
\Gamma
. Various properties of generalized Lie groups
G
G
and their locally convex topological Lie algebras
L
G
=
H
o
m
(
R
,
G
)
LG = Hom({\mathbf {R}},G)
are investigated.