Results 1 to 2 of 2

Thread: Metric space of bounded real functions is separable iff the space is finite. Prove.

  1. #1
    Junior Member
    Joined
    Sep 2011
    Posts
    30

    Post Metric space of bounded real functions is separable iff the space is finite. Prove.

    A metric space, B(S), with the "uniform metric" d(f,g) = sup {|f - g| }, of bounded, real valued functions on the set S is separable if and only if the space is finite. This is ex. 7 in the "Introduction to Topology" by Gamelin and Green (Second Edition) Part I. Section 5, which I am reading to keep my retired brain cells active. Regrettably, they are not active enough, because the hint supplied by the authors has not helped me. The hint defines a characteristic function for subsets of S as follows:
    Chi-subT (s) = 1 for s belongs to T
    Chi-subT (s) = 0 for s belongs to S-T
    With this definition the open balls B(Chi-subT; 1/2) are disjoint, the authors state. That is the hint.

    I see from the metric that the balls are indeed disjoint. I have not been able to use this property to show that S is finite if there is a countably dense subset of functions in B(S), nor given that the set S is finite that there is a countably dense set in B(S). I surmise that perhaps the zeros of a polynomial with real polynomial of degree N is involved in showing that S is finite; also that the polynomial is obtained from the limit that d(f, h-subj) = 0 where f is any bounded function in B(S) and {h-subj} is a countable set dense in B(S). I have not been able to carry through the argument, sadly.

    Can anyone help?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Sep 2011
    Posts
    30

    Re: Metric space of bounded real functions is separable iff the space is finite. Prov

    Please ignore this posting. I have re-submitted it with the same title using LaTex for easier reading.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: Oct 9th 2013, 12:02 PM
  2. Replies: 2
    Last Post: Jul 8th 2011, 03:16 PM
  3. Open sets in a separable metric space
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Sep 24th 2009, 02:03 PM
  4. Replies: 1
    Last Post: Feb 8th 2009, 04:12 PM
  5. [SOLVED] is space of bounded continuous functions separable?
    Posted in the Advanced Math Topics Forum
    Replies: 9
    Last Post: Jan 18th 2009, 09:16 PM

/mathhelpforum @mathhelpforum