Let A ⊂ X {\displaystyle A\subset X} , and ( X , d ) {\displaystyle (X,d)} a metric space.
We denote i n t ( A ) = { x ∈ X : ∃ ϵ > 0 , B ( x , ϵ ) ⊂ A } {\displaystyle int(A)=\{x\in X:\exists \epsilon >0,B(x,\epsilon )\subset A\}}
We denote e x t ( A ) = { x ∈ X : ∃ ϵ > 0 , B ( x , ϵ ) ⊂ X ∖ A } {\displaystyle ext(A)=\{x\in X:\exists \epsilon >0,B(x,\epsilon )\subset X\backslash A\}}
Finally we denote b r ( A ) = { x ∈ X : ∀ ϵ > 0 , ∃ y , z ∈ B ( x , ϵ ) , y ∈ A , z ∈ X ∖ A } {\displaystyle br(A)=\{x\in X:\forall \epsilon >0,\exists y,z\in B(x,\epsilon ),{\text{ }}y\in A,z\in X\backslash A\}}
Let A ⊂ X {\displaystyle A\subset X} , and ( X , d ) {\displaystyle (X,d)} be a metric space.
i n t ( A ) ∪ b r ( A ) ∪ e x t ( A ) = X {\displaystyle int(A)\cup br(A)\cup ext(A)=X}
i n t ( A ) {\displaystyle int(A)} , b r ( A ) {\displaystyle br(A)} , and e x t ( A ) {\displaystyle ext(A)} are disjoint.
We denote c l ( A ) = A ∪ L i m ( A ) {\displaystyle cl(A)=A\cup Lim(A)}
c l ( A ) = A ∪ b r ( A ) {\displaystyle cl(A)=A\cup br(A)}
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