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Math Help - Hausdroff

  1. #1
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    Hausdroff

    If  x,y are two distinct points in a metric space  X then there exists  r > 0 such that  B_{r}(x) and  B_{r}(y) are disjoint.

    So let  p = d(x,y) > 0 . Choose  r = p/2 . Then  B_{r}(x) and  B_{r}(y) are disjoint for all  l \leq r ?

    Is this right? e.g.  r = \sup A where  A = \{l:  B_{r}(l) \ \text{and} \ B_{l}(y) \ \text{are disjoint} \} ?
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  2. #2
    Moo
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    Quote Originally Posted by particlejohn View Post
    If  x,y are two distinct points in a metric space  X then there exists  r > 0 such that  B_{r}(x) and  B_{r}(y) are disjoint.

    So let  p = d(x,y) > 0 . Choose  r = p/2 . Then  B_{\color{red}l}(x) and  B_{\color{red}l}(y) are disjoint for all  l \leq r ?
    If they're open balls, yep.

    Is this right? e.g.  r = \sup A where  A = \{l:  B_{r}(l) \ \text{and} \ B_{l}(y) \ \text{are disjoint} \} ?
    Isn't there a problem with the indices?
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