On the structure of generalized rough sets

In this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that

1.

Any reflexive binary relation determines a topology.

2.

If θ is a reflexive and symmetric relation on a set X, then O={A⊆X|θ_-(A)=A} is a topology such that A is open if and only if it is closed.

3.

Conversely, for every topological space (X,O) satisfying the condition that A is open if and only if it is closed, there exists a reflexive and symmetric relation R such that O={A⊆X|R_-(A)=A}.

4.

Let θ be an equivalence relation on X. For any pseudo ω-closed subset A of X, θ₋(A) is an ω-closed set if and only if ω(x, x, … , x) ∈ θ₋(A) for any x ∈ X.

Moreover we consider properties of generalized rough sets.