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Consider R² with the usual topology and A,B subset or equal R² then check weather int(A union B)=int(A) union int(B) or not?
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Consider R² with the usual topology and A,B subset or equal R² then ch...
Answer:
Intuition:
The interior of a set consists of all points that are contained within the set without touching the boundary. When we take the union of two sets, the interior of the union may not necessarily be the union of the interiors of the individual sets.
Counterexample:
Let's consider a counterexample to show that int(A ∪ B) ≠ int(A) ∪ int(B).
- Let A = {(x, y) ∈ R² : x² + y² < 1}="" be="" the="" open="" unit="" disk="" centered="" at="" the="" origin.="" -="" let="" b="{(x," y)="" ∈="" r²="" :="" (x-2)²="" +="" y²="" />< 1}="" be="" the="" open="" unit="" disk="" centered="" at="" (2,="" 0).="" then,="" int(a)="A" and="" int(b)="B." however,="" int(a="" ∪="" b)="" contains="" points="" both="" inside="" the="" disk="" centered="" at="" the="" origin="" and="" inside="" the="" disk="" centered="" at="" (2,="" 0),="" which="" is="" not="" equal="" to="" the="" union="" of="" int(a)="" and="" int(b).="" therefore,="" int(a="" ∪="" b)="" ≠="" int(a)="" ∪="" int(b)="" in="" general.="" />
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Consider R² with the usual topology and A,B subset or equal R² then check weather int(A union B)=int(A) union int(B) or not?
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Consider R² with the usual topology and A,B subset or equal R² then check weather int(A union B)=int(A) union int(B) or not? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Consider R² with the usual topology and A,B subset or equal R² then check weather int(A union B)=int(A) union int(B) or not? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider R² with the usual topology and A,B subset or equal R² then check weather int(A union B)=int(A) union int(B) or not?.
Consider R² with the usual topology and A,B subset or equal R² then check weather int(A union B)=int(A) union int(B) or not? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Consider R² with the usual topology and A,B subset or equal R² then check weather int(A union B)=int(A) union int(B) or not? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider R² with the usual topology and A,B subset or equal R² then check weather int(A union B)=int(A) union int(B) or not?.
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