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Unformatted text preview: X . (3) If y ∈ D ( x,r ), then D ( x,r ) = D ( y,r ). (4) If D ( x,r 1 ) ∩ D ( y,r 2 ) 6 = ∅ , then either D ( x,r 1 ) ⊂ D ( y,r 2 ) or D ( y,r 2 ) ⊂ D ( x,r 1 ). 5. [20 points] Let [ a,b ] be an interval and X = C ([ a,b ]) the vector space of all continuous realvalued functions on [ a,b ] equipped with the norm k f k = max a ≤ t ≤ b  f ( t )  . Let F be a ﬁxed continuous function such that F ( t ) > 0 for all t ∈ [ a,b ] and let S = { f ∈ X :  f ( t )  < F ( t ) for all t ∈ [ a,b ] } . 1 (1) Prove that S is open. (2) Find cl( S ). (3) Find ∂S . Remark. If you have diﬃculties solving this problem you should attend the tutorial (the problem will be solved there). 2...
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 Spring '09
 Math, Topology, Metric space, Topological space, continuous realvalued functions

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