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hw5sol - 147 HW5 Solutions 1(19.6 Let x1 x2 be a sequence...

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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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