Hi I've been struggling with this problem and I would really appreciate any help or hints anybody could give me.
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Let (X,d) and (Y,p) be metric spaces and
f:X->Y is a continuous function
Suppose C is a compact subset of X such that:
the restriction f|_C: C->Y is injective and for each x in C there exists a real number r(x)>0 such that, on the open ball (neighborhood) with radius r(x) around x consists of all x' in X such that d(x,x')
i.e. B(x,r(x))={x' in X|d(x,x')
and the restriction f|_B(x,r(x)):B(x,r(x))->Y is injective.
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prove that there exists a real number R>0 such that on the open subset G of X where G consists of the Union of all open balls of radius R centered at x
[i.e. B(x,R)] for all x in C
G=U_(x in C) B(x,R)
That the restriction f|_G: G->Y is injective
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Thank you so much for taking the time to help me
2007-11-27
17:19:54
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1 answers
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asked by
greeneggs4spam
3
in
Mathematics