In this paper, we present the covering rough sets based on neighborhoods by approximation operations as a new type of extended covering rough set models. In fact, we have introduced
generalizations to W. Zhu approaches (Zhu, 2007). Based on the notion of neighborhood induced
from any binary relation, four different pairs of dual approximation operators are defined with their
properties being discussed. The relationships among these operators are investigated. Finally, an
interesting theorem to generate different topologies is provided. Comparisons between these
topologies are discussed. In addition, several examples and counter examples to indicate counter
connections are investigated.