hw5sol - 147 HW5 Solutions 1) (19.6) Let x 1 , x 2 ,... be...

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Unformatted text preview: 147 HW5 Solutions 1) (19.6) Let x 1 , x 2 ,... be a sequence of the points of the product space Q X α . Show that this sequence converges to the point x if and only if the sequence π α ( x 1 ) ,π α ( x 2 ) ,... converges to π α ( x ) for each α . Is this fact true if one uses the box topology instead of the product topology? Solution: Suppose π α ( x 1 ) ,π α ( x 2 ) ,... converges to π α ( x ) for each α . Let V be a neighborhood of x , then as sets of the form π- 1 α ( U ) give a subbasis for the product topology, V is the union of finite intersections of such sets. Then x is contained in some finite intersection T α i π- 1 α i ( U α i ) ⊂ V . For each U α i , let n i be some large integer such that π ( x k ) ∈ U α i for all k > n i . Then for all k > max { n i } , x k ∈ T α i π- 1 α i ( π α i ( x k )) ⊂ T α i π- 1 α i ( U α i ) ⊂ V . So we have that x 1 , x 2 ,... converges to x . Suppose x 1 , x 2 ,... converges to x . Pick some arbitrary α . For any neigborhood U of π α ( x ), π- 1 α ( U ) contains all x k for k greater than some n . Thus π α ( x k ) ∈ U for all k greater than the same n ....
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This note was uploaded on 05/19/2010 for the course MAT mat147 taught by Professor Derekwise during the Spring '10 term at UC Davis.

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hw5sol - 147 HW5 Solutions 1) (19.6) Let x 1 , x 2 ,... be...

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