Results 1 to 2 of 2

Math Help - [SOLVED] Metric Space- Distance Function

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    4

    Exclamation [SOLVED] Metric Space- Distance Function

    Hey guys, I have a problem from my Topology course that I can't seem to figure out. Any help would be appreciated. Here it is:

    The distance functions D'=D/(1+D) and D''= min{D,1} are each metrics on M. Let x \inM and let \epsilon \in R>0.
    Let:

    S \epsilon'(x)={y \inM|D'(x,y)< \epsilon}
    S \epsilon''(x)={y \inM|D''(x,y)< \epsilon}

    a) Prove that S \epsilon''(x) \subseteq S \epsilon]'(x).
    b)Prove that S (\epsilon/(1+\epsilon))'(x) \subseteq S \epsilon''(x).

    It almost seems to me like the containment should be switched in part a at least, but I don't know.
    Thanks in Advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by tidus89 View Post
    Hey guys, I have a problem from my Topology course that I can't seem to figure out. Any help would be appreciated. Here it is:

    The distance functions D'=D/(1+D) and D''= min{D,1} are each metrics on M. Let x \inM and let \epsilon \in R>0.
    Let:

    S \epsilon'(x)={y \inM|D'(x,y)< \epsilon}
    S \epsilon''(x)={y \inM|D''(x,y)< \epsilon}

    a) Prove that S \epsilon''(x) \subseteq S \epsilon]'(x).
    b)Prove that S (\epsilon/(1+\epsilon))'(x) \subseteq S \epsilon''(x).

    It almost seems to me like the containment should be switched in part a at least, but I don't know.
    Thanks in Advance.
    This notation is hard to understand (at least for me), but I would tend to agree. If D>1 then \frac{D}{1+D}=1-\frac{1}{1+D}<1=\min\left(D,1\right)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: November 23rd 2011, 12:54 AM
  2. Function from Metric Space to R is Continuous
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: February 28th 2011, 11:29 AM
  3. Limit of function from one metric space to another metric space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 17th 2010, 03:04 PM
  4. [SOLVED] Metric Space basic definition
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: September 4th 2008, 05:11 AM
  5. [SOLVED] Metric Space and Inequality
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 31st 2008, 05:08 PM

Search Tags


/mathhelpforum @mathhelpforum