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OPc0bM-08

# OPc0bM-08 - ASSIGNMENT 8 SOLUTIONS MAT 473 SPRING 2012...

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ASSIGNMENT 8 · SOLUTIONS MAT 473 · SPRING 2012 Exercise 14.1. For A R n and x R n , prove that A + x L if and only if A L , in which case λ ( A + x ) = λ ( A ). Proof. For any x R n , translation by x is continuous on R n with continuous inverse (namely, translation by x ). It follows that a set G is open if and only if G + x is open. Also, recall from Exercise 12.2 that λ ( G + x ) = λ ( G ) for all open G R n . Now consider any set A R n . Since a set G is open with A + x G if and only if G x is open with A G x , it follows that λ ( A + x ) = inf { λ ( G ) | G is open and A + x G } = inf { λ ( G ) | G x is open and A G x } = inf { λ ( G x ) | G x is open and A G x } = inf { λ ( H ) | H is open and A H } = λ ( A ) . Since a set K is compact if and only if K + x is compact, it follows from this that λ ( K + x ) = λ ( K ) for all compact K . Arguing as above, for any A R n we now have λ ( A + x ) = sup { λ ( K ) | K is compact and K A + x } = sup { λ ( K ) | K x is compact and K x A } = sup { λ ( K x ) | K x is compact and K

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OPc0bM-08 - ASSIGNMENT 8 SOLUTIONS MAT 473 SPRING 2012...

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