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Homework Help: 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)

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