1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Which of the following topologies are metrizable?

  1. Mar 21, 2012 #1
    Can someone please help me to determine which of the following topologies are metrizable?

    a)Let X be any inf nite set and let T = {U subset of X : x\u is finite }

    b) Let X = R and let T = {U subset of R : R \ U is FI nite of countable }

    c) For each k in N, let Nk = {1; 2,....,K} . Let T = {empty set} U{N}U {Nk: k is in N}

    d) For each k in N, let U = { k; k + 1; k + 2;....} then T = { empty set} U {Uk : k is in N}

    e) Let T = {empty set} U { R} U { (a, infinity) : a is in R}


    I think that {e} is not metrizable (weak topology and not countable)
    Also, I think that a) and b) are metrizable , however not sure about c) and d)

    thanks
     
  2. jcsd
  3. Mar 21, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Can you list some properties of metrizable spaces??

    For example, I know that in a metrizable space

    1) Every singleton is closed
    2) It is first countable
    3) ...

    Can you list some more properties??
     
  4. Mar 21, 2012 #3
    they are always Hausdorff ?

    Am I right for e) a) and b) ?
     
  5. Mar 21, 2012 #4
    Actually, I think a) is not metrizable because it is not Hausedorff ?

    Also, for b) the cocountable topology is not Hausedorff therefore b) is not metrizable?
     
    Last edited: Mar 21, 2012
  6. Mar 21, 2012 #5
    You are right. That finite complement topology is the first non-metrizable space I ever know. Similar for (b)
     
    Last edited: Mar 21, 2012
  7. Mar 21, 2012 #6
    thanks, what do you think about b) please see the post above yours
     
  8. Mar 21, 2012 #7
    I think your reasoning is right, since R is uncountable.
     
  9. Mar 21, 2012 #8

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Is e Hausdorff??
     
  10. Mar 21, 2012 #9
    Well, before I thought it wasn't, but I am not sure anymore....it seems like it is...

    c) and d) are Hausdorff, therefore metrizable, right?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Which of the following topologies are metrizable?
Loading...