Finite complement topology

  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. morphism

    morphism 2,020
    Science Advisor
    Homework Helper

    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. 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. 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. Topology

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

    Moonbear 11,955
    Staff Emeritus
    Science Advisor
    Gold Member

    Note: I have merged your two threads on this since you had already received responses in the Calc and Analysis forum.
  8. Hello?
  9. HallsofIvy

    HallsofIvy 41,265
    Staff Emeritus
    Science Advisor

    Look at morphism's post!
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