Results 1 to 4 of 4

Math Help - Metric space subset

  1. #1
    Newbie
    Joined
    Oct 2012
    From
    NJ
    Posts
    2

    Metric space subset

    This is a homework problem in my algebra analysis class. I'm not really sure where to begin. I need to prove the statement below.

    Y X where X is a metric space with function d. Then (Y,d) is a metric space with the same function d. How can I show that X and Y have the same metric function?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,417
    Thanks
    718

    Re: Metric space subset

    Quote Originally Posted by selzer9 View Post
    Y X where X is a metric space with function d. Then (Y,d) is a metric space with the same function d. How can I show that X and Y have the same metric function?
    You can't show this. To show this, you to need consider the metric function of X and the metric function of Y. You can only talk about a metric function of a metric space. But, if I understand the problem correctly, Y is not a metric space by definition; it's just a subset of X. You need to show that this subset together with the metric function of X forms a metric space. And to do this, check the axioms of a metric space.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,390
    Thanks
    1476
    Awards
    1

    Re: Metric space subset

    Quote Originally Posted by selzer9 View Post
    Y\subset X where X is a metric space with function d. Then (Y,d) is a metric space with the same function d. How can I show that X and Y have the same metric function?
    Assuming that (X,d) is a metric space and Y\subset X then (Y,d) is a metric space. If you study this definition of a metric, then you see that the function d does not loose any of its properties when applied to points of Y.
    Last edited by Plato; October 2nd 2012 at 03:08 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,150
    Thanks
    591

    Re: Metric space subset

    strictly speaking you should speak of a separate function d*, the restriction of d to YxY. this is because (in some views) the domain and co-domain of a function are part of the function's definition. in other words, it's not just "the rule" that defines a function, it's also "what you apply the rule TO", and "where the values of the function LIVE".

    but...it is common practice to make no distinction between the surjective function

    f:A→f(A)

    and the "general function" it came from:

    f:A→B. be aware of this...it can trip you up sometimes.

    of course d* "is really" just d, because:

    d*(y1,y2) = d(y1,y2), where on the right, we are considering the y's as elements of X.

    in general, any statement of the form:

    FOR ALL x,y,z in X: property (insert something here) is true, remains true even if we pick x,y and z from a SUBSET of X.

    ******

    informally speaking, if two points lie within "an epsilon ball" in X, and both points are in Y, then they are in "an epsilon ball" in Y (but: "epsilon balls" in Y aren't always "round", parts of them might be "chopped off" because they lie in the part of X outside Y).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: September 26th 2012, 09:06 AM
  2. Replies: 2
    Last Post: July 8th 2011, 02:16 PM
  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, 02:04 PM
  4. A finite subset of a metric space
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 10th 2009, 05:48 PM
  5. Replies: 5
    Last Post: September 9th 2007, 01:57 PM

Search Tags


/mathhelpforum @mathhelpforum