INTERIOR OF A SET, DERIVED AND CLOSURE OF A SET #interior #derived #closure
https://youtu.be/DSsOiyrWjfo?si=pjwdN8c9pz5Ca2WT
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 finite set) =ⲫ
* D(N) = D(Z) = ⲫ
* D(Q) = D(R) = R
* If A= (a, b) or A = [a, b] or A = [a, b) or A = (a, b] then D(A) = [a, b].
* D(A)⊂ A
Closure of a set:-
The closure of a non-empty set A in R is the set of all limit points of A .
in other words, closure of A is an union of A and it's derived set D(A) and closure of A is denoted by Δ.
∴ Δ = A∪ D(A)
Properties of closure of A:-
* If A=[a, b] then Δ = [a, b]
* If A=(a, b) then Δ=[a, b]
* If A= ⲫ then Δ=R
* If A=R then Δ = R
* Δ is a closed set
* We always have A⊂ Δ
* A is closed iff A = Δ
* Since D(N)=ⲫ, closure(N)=N∪D(N)=N ∪ ⲫ = N.
∴ N is closed.
Similarly Z is closed.
* Since D(Q)= R , closure (Q)= Q∪D(Q)=Q∪R=R
i.e. closure(Q) =R≠ Q
∴ Q is not closed.
people also ask
what is a neighborhood of a point?
what are subsets of the real line R?
#interior #derived #closure #limit #point
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