  In this paper we adhere to the definition of infra-topological space as it
was introduced by Al-Odhari. Namely, we speak about families of subsets which
contain empty set and the whole universe X, being at the same time closed under
finite intersections (but not necessarily under arbitrary or even finite
unions). This slight modification allows us to distinguish between new classes
of subsets (infra-open, ps-infra-open and i-genuine). Analogous notions are
discussed in the language of closures. The class of minimal infra-open sets is
studied too, as well as the idea of generalized infra-spaces. Finally, we
obtain characterization of infra-spaces in terms of modal logic, using some of
the notions introduced above.
