  We introduce a new class of infinite-dimensional Lie algebras, which we refer
to as continuum Kac-Moody algebras. Their construction is closely related to
that of usual Kac-Moody algebras, but they feature a continuum root system with
no simple roots. Their Cartan datum encodes the topology of a one-dimensional
real space and can be thought of as a generalization of a quiver, where
vertices are replaced by connected intervals. For these Lie algebras, we prove
an analogue of the Gabber-Kac-Serre theorem, providing a complete set of
defining relations featuring only quadratic Serre relations. Moreover, we
provide an alternative realization as continuum colimits of symmetric
Borcherds-Kac-Moody algebras with at most isotropic simple roots. The approach
we follow deeply relies on the more general notion of a semigroup Lie algebra
and its structural properties.
