S16 - 1st December 2004 Munkres 16 Ex. 16.1 (Morten...

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1st December 2004 Munkres § 16 Ex. 16.1 (Morten Poulsen). Let ( X, T ) be a topological space, ( Y, T Y ) be a subspace and let A Y . Let T Y A be the subspace topology on A as a subset of Y and let T X A be the subspace topology on A as a subset of X . Since U ∈ T Y A ⇔ ∃ U Y ∈ T Y : U = A U Y ⇔ ∃ U X ∈ T : U = A ( Y U X ) ⇔ ∃ U X ∈ T : U = A U X U ∈ T X A it follows that T Y A = T X A . Ex. 16.3 (Morten Poulsen). Consider Y = [ - 1 , 1] as a subspace of R with the standard topology. By lemma 16.1 a basis for the subspace topology on Y is sets of the form: Y ( a, b ) = ( a, b ) , a, b Y [ - 1 , b ) , a / Y, b Y ( a, 1] , a Y, b / Y Y, , a, b / Y. Note that intervals of the form [ a, b ) are not open in R , since there are no basis element ( c, d ) such that a ( c, d ) [ a, b ). Similarly are intervals of the form ( a, b ] and [ a, b ] not open in R . A = { x | 1 2 < | x | < 1 } : A = ( - 1 , - 1 2 ) ( 1 2 , 1), hence A open in R . Since A = Y A it follows that A open in Y .
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S16 - 1st December 2004 Munkres 16 Ex. 16.1 (Morten...

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