This paper puts forward some rough approximations which are motivated from topology. Given a subset R ⊆ U × U, we can use 8 types of E-neighborhoods to construct approximations of an arbitrary X⊆ U on the one hand. On the other hand, we can also
construct approximations relying on a topology which is induced by an E-neighborhood. Properties of these approximations and relationships between them are studied. For convenience of use, we also give some useful and easy-to-understand examples and
make a comparison between our approximations and those in the published literature.