Abstract
We study a topology of the class of proper submodules of a module and
some of its distinguished subclasses, which we call submodule spaces. We
provide several criteria for the quasi-compactness of these submodule spaces.
Additionally, we examine the T0 and T1 separation properties and character-
ize submodule spaces where nonempty irreducible closed subsets have unique
generic points. We give a sufficient condition for the connectedness of sub-
module spaces. Furthermore, we prove that the submodule spaces of proper
submodules are spectral and characterize the spectral submodule spaces of
Noetherian modules. Finally, we discuss continuous functions between these
spaces.