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Difference between open sphere and epsilon-neighbourhood - Metric Spaces

  1. Jul 8, 2012 #1
    In Elements of the Theory of Functions and Functional Analysis (Kolmogorov and Fomin) the definitions are as follows:

    An open sphere [itex]S(x_0,r)[/itex] in a metric space [itex]R[/itex] (with metric function [itex]\rho(x,y)[/itex]) is the set of all points [itex]x\in R[/itex] satisfying [itex]\rho(x,x_0)<r[/itex]. The fixed point [itex]x_0[/itex] is called the center; the number [itex]r[/itex] is called the radius.

    An ε-neighbourhood of the point [itex]x[/itex], denoted [itex]O(x,\epsilon)[/itex], is an open sphere of radius ε and center [itex]x_0[/itex].

    How is the ε-neighbourhood a significant definition? It seems to be just the open sphere with a different radius symbol. If we have a neighbourhood of a point [itex]x[/itex] (as per the definition) does this [itex]x[/itex] have to lie within the open sphere? I mean, I assume it must, but this doesn't seem to be captured by the definition. What am I missing?

    Thanks.
     
  2. jcsd
  3. Jul 8, 2012 #2

    HallsofIvy

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    No. an [itex]\epsilon-neighborhood[/itex] of the point [itex]x[/itex] is open sphere with radius [itex]\epsilon[/itex] and center x, not some [itex]x_0[/itex] (that wouldn't make sense because [itex]x_0[/itex] is not mentioned in the notation "[itex]O(x, \epsilon)[/itex]").

    An "[itex]\epsilon[/itex]-neighborhood of x" is specifically the open ball of radius [itex]\epsilon[/itex] with radius [itex]\epsilon[/itex] that is centered at x.
     
  4. Jul 8, 2012 #3
    I think this should read

    An ε-neighbourhood of the point x, denoted O(x,ϵ), is an open sphere of radius ε and center x.

    I have bolded the difference.

    I prefer the term open ball to open sphere however.

    edit I see Halls of ivy just beat me.

    :biggrin:
     
  5. Jul 8, 2012 #4

    HallsofIvy

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    Yahoo!:tongue:
     
  6. Jul 9, 2012 #5
    That's as I suspected. There must be a typo in the book.

    Suddenly it all makes sense. :smile:
     
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