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Unformatted text preview: f ( M ) is actually closed. To see this, let ( f ( a n ) : n N ) be a sequence in f ( M ) converging to b in M . From d ( a q ,a p ) = d ( f ( a q ) ,f ( a p )) it follows that a n : n N ) is Cauchy, hence (as compact implies complete) convergent, say a n a ; from d ( f ( a ) ,f ( a n )) = d ( a,a n ) it follows that f ( a n ) f ( a ). Finally, uniqueness of limits forces b = f ( a ) and places b in f ( M ). Remark : The notion of continuity simplies some of this argument: f ( M ) is the image of a compact space under a continuous map, thus compact and so closed. 1...
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This note was uploaded on 11/09/2011 for the course MAA 5228 taught by Professor Robinson during the Fall '09 term at University of Florida.
- Fall '09