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Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=interior(A) union interior(B) or not?
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Consider R² with the usual topology and A,B subset or equal R² then ch...
Proof:
Interior of Union vs Union of Interiors
Interior of (A union B) is not equal to interior of A union interior of B in general.
Counterexample:
Consider A = {(x, y) ∈ R² : x < 1}="" and="" b="{(x," y)="" ∈="" r²="" :="" y="" />< 1}.="" />
- The interior of A is the open set {(x, y) ∈ R² : x < 1}="" which="" is="" equivalent="" to="" the="" open="" interval="" (-∞,="" 1)="" ×="" r="" in="" />
- The interior of B is the open set {(x, y) ∈ R² : y < 1}="" which="" is="" equivalent="" to="" the="" open="" interval="" r="" ×="" (-∞,="" 1)="" in="" />
- The union of A and B is the set {(x, y) ∈ R² : x < 1="" or="" y="" />< 1}="" which="" has="" interior="" equal="" to="" the="" open="" set="" {(x,="" y)="" ∈="" r²="" :="" x="" />< 1="" or="" y="" />< />
- The union of the interiors of A and B would be the union of the open sets (-∞, 1) × R and R × (-∞, 1), which is the entire R².
Therefore, in this case, the interior of (A union B) is not equal to the union of the interiors of A and B.
Interior of Union vs Union of Interiors
Interior of (A union B) is not equal to interior of A union interior of B in general.
Counterexample:
Consider A = {(x, y) ∈ R² : x < 1}="" and="" b="{(x," y)="" ∈="" r²="" :="" y="" />< 1}.="" />
- The interior of A is the open set {(x, y) ∈ R² : x < 1}="" which="" is="" equivalent="" to="" the="" open="" interval="" (-∞,="" 1)="" ×="" r="" in="" />
- The interior of B is the open set {(x, y) ∈ R² : y < 1}="" which="" is="" equivalent="" to="" the="" open="" interval="" r="" ×="" (-∞,="" 1)="" in="" />
- The union of A and B is the set {(x, y) ∈ R² : x < 1="" or="" y="" />< 1}="" which="" has="" interior="" equal="" to="" the="" open="" set="" {(x,="" y)="" ∈="" r²="" :="" x="" />< 1="" or="" y="" />< />
- The union of the interiors of A and B would be the union of the open sets (-∞, 1) × R and R × (-∞, 1), which is the entire R².
Therefore, in this case, the interior of (A union B) is not equal to the union of the interiors of A and B.
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Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=interior(A) union interior(B) or not?
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Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=interior(A) union interior(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 interior(A union B)=interior(A) union interior(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 interior(A union B)=interior(A) union interior(B) or not?.
Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=interior(A) union interior(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 interior(A union B)=interior(A) union interior(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 interior(A union B)=interior(A) union interior(B) or not?.
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