Rough set theory is one of important models of granular computing. Lower and
upper approximation operators are two important basic concepts in rough set
theory. The classical Pawlak approximation operators are based on partition
and have been extended to covering approximation operators. Covering is one
of the fundamental concepts in the topological theory, then topological
methods are useful for studying the properties of covering approximation
operators. This paper presents topological properties of a type of granular
based covering approximation operators, which contains seven pairs of
approximation operators. Then, topologies are induced naturally by the seven
pairs of covering approximation operators, and the topologies are just the
families of all definable subsets about the covering approximation
operators. Binary relations are defined from the covering to present
topological properties of the topological spaces, which are proved to be
equivalence relations. Moreover, connectedness, countability, separation
property and Lindel?f property of the topological spaces are discussed. The
results are not only beneficial to obtain more properties of the pairs of
covering approximation operators, but also have theoretical and actual
significance to general topology.
$\mathcal F$-hypercyclic and disjoint $\mathcal F$-hypercyclic properties of binary relations over topological spaces