Unformatted text preview: C 6 f ( b + a ); as b + a 1 + b + a = b 1 + b + a + a 1 + b + a 6 b 1 + b + a 1 + a we deduce that C 6 B + A as required. (ii) Let p,x ∈ M and δ > 0. As f is strictly increasing, d ( x,p ) < δ ⇔ D ( x,p ) < f ( δ ) =: Δ so that B d δ ( p ) = B D Δ ( p ) . Thus d-balls and D-balls (of radius less than unity) are precisely the same sets, and so d-open and D-open have precisely the same meaning. Moral : In dealing with properties of metric spaces that may be formulated in terms of open sets, it may be assumed that the metric is bounded. 1...
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- Fall '09
- Metric space, Modern Analysis