  In this paper, we have established boundaries of cardinal numbers of nonempty
sets in finite non-$T_1$ topological spaces using interval analysis. For a
finite set with known cardinality, we give interval estimations based on the
closure and interior of the set. In this paper, we give new results for the
cardinalities of non-empty semi-open sets in non-$T_1$ topological spaces as
well as in extremely disconnected and hyperconnected topological spaces.
