INTERIOR OF A SET, DERIVED AND CLOSURE OF A SET #interior #derived #closure
In this session we see the definitions and some properties of interior of a set, derived set and closure of a set. Interior of a set :- The interior of a subset A in R is the set of all interior points of A and it is denoted by Int(A). i.e. Int(A) = the set of all interior points of A Properties :- * Int(A) is the union of all open subsets of A. * Int(A) is always open. * Int(A) ⊆ A * Int(A)=A iff A is open. * Int(Int(A)) = Int(A) * Int(R) = R and Int(ⲫ) = ⲫ * Int(A∩B) = Int(A) ∩ Int(B) * Int(A) is the largest open subset of A. Examples :- 1. Int(a, b) = (a, b) 2. Int[a, b] = (a, b) 3. Int(Z) =ⲫ ; Int (N) = ⲫ ; Int(Q) = ⲫ 4. Int(a finite set) = ⲫ Derived set :- The set of all limit points of a non-empty set A of R is called the derived set of A and is denoted by D(A). Examples :- * Since empty set has no limit points, D(ⲫ)=ⲫ * D( a...