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Homework Help: Finite complement topology

  1. Oct 26, 2007 #1
    I need to show if the finite complement topology,T_3, and the topology having all sets (-inf,a) = {x|x<a} as basis ,T_5, are comparable.

    I've shown that T_3 is not strictly finer than T_5.

    But I'm not sure about other case.

    I need help.
    Last edited by a moderator: Mar 8, 2013
  2. jcsd
  3. Oct 27, 2007 #2


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    I presume you're defining these topologies on R and that you managed to find a set in T_5 that's not in T_3. The other direction is just as easy: is R\{0} in T_5?
  4. Oct 27, 2007 #3
    R\{0} is not in T_5. (-inf,0]U[0,inf)

    If T_4 is the upper limit topology, having the sets (a,b] as a basis and
    T_2 the topology of R_K (a,b)-K K = 1/n n in Z.

    I've shown that T_2 is not strictly finer than T_4.
    How do I show that T_4 is strictly finer than T_2?
    Last edited: Oct 27, 2007
  5. Oct 27, 2007 #4
    And most importantly, why is R\{1,2,3} considered a basis element for T_3?

    It's not finite, nor is it all of R. I'm confused with the definition of a finite complement topology.
  6. Oct 27, 2007 #5

    Is {1} = (0,2) ?
    and R\{0} = (-inf,0]U[0,inf) ?
    Last edited: Oct 27, 2007
  7. Oct 27, 2007 #6


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    Note: I have merged your two threads on this since you had already received responses in the Calc and Analysis forum.
  8. Oct 28, 2007 #7
  9. Oct 29, 2007 #8


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    Look at morphism's post!
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