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β1 -paracompact spaces

“…α-paracompact [5] (resp., P 1 -paracompact [15], S 1 -paracompact [3]) spaces are defined by replacing the open cover in original definition by α-open (resp., preopen, semiopen) cover. A subset A of space X is said to be N -closed relative to X (briefly, N -closed) [9] if for every cover {U α : α ∈ ∆} of A by open sets of X, there exists a finite subfamily ∆ • of ∆ such that A ⊂ {Int(Cl(U α )) : α ∈ ∆ • }.…”

“…A space (X, τ ) is said to be α-paracompact [5] (resp.,P 1 -paracompact [15], S 1 -paracompact [3]), if every α-open (resp., preopen, semiopen) cover of X has a locally finite open refinement.…”

Certain approximation spaces using local functions via idealization

“…The generated topology τ on an approximation space (X, R) that represent the topological properties of rough sets were studied by many authors (ex. [2,[6][7][8][9][10]). Many kinds of generalizations of Pawlak's rough set were obtained by replacing the equivalence relation with an arbitrary binary relation.…”