  We consider some distinguished classes of elements of a multiplicative
lattice endowed with coarse lower topologies, and call them lower spaces. The
primary objective of this paper is to study the topological properties of these
lower spaces, encompassing lower separation axioms and compactness. We
characterize lower spaces that exhibit sobriety. Introducing the concept of
strongly disconnected spaces, we establish a correlation between strongly
disconnected lower spaces and the presence of nontrivial idempotent elements in
the corresponding multiplicative lattices. Additionally, we provide a
sufficient condition for a lower space to be connected. We prove that the lower
space of proper elements is a spectral space, and we further explore continuous
maps between lower spaces.
