Topological characterizations to three types of covering approximation
  operators
  Covering-based rough set theory is a useful tool to deal with inexact,
uncertain or vague knowledge in information systems. Topology, one of the most
important subjects in mathematics, provides mathematical tools and interesting
topics in studying information systems and rough sets. In this paper, we
present the topological characterizations to three types of covering
approximation operators. First, we study the properties of topology induced by
the sixth type of covering lower approximation operator. Second, some
topological characterizations to the covering lower approximation operator to
be an interior operator are established. We find that the topologies induced by
this operator and by the sixth type of covering lower approximation operator
are the same. Third, we study the conditions which make the first type of
covering upper approximation operator be a closure operator, and find that the
topology induced by the operator is the same as the topology induced by the
fifth type of covering upper approximation operator. Forth, the conditions of
the second type of covering upper approximation operator to be a closure
operator and the properties of topology induced by it are established. Finally,
these three topologies space are compared. In a word, topology provides a
useful method to study the covering-based rough sets.
