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Homework Help: Finite-Compliment Topology and intersection of interior

  1. Oct 2, 2007 #1
    [SOLVED]Finite-Compliment Topology and intersection of interior

    1. The problem statement, all variables and given/known data

    Given topological space (R[tex]^{1}[/tex], finite compliment topology), find counter example to show that

    Arbitary Intersection of (interior of subset of R[tex]^{1}[/tex]) is not equal to Interior of (arbitary intersection of subset of R[tex]^{1}[/tex]).

    [tex]\bigcap^{\infty}_{n=1}int(A_{n})\neq int(\bigcap^{\infty}_{n=1}A_{n}) [/tex]

    2. Relevant equations

    When we consider topological space (R[tex]^{1}[/tex], usual topology), it is easy to find out that [-1/n, 1/n] is the example.

    3. The attempt at a solution

    First, I thought what shapes the open of topological space (R[tex]^{1}[/tex], finite compliment topology) might have, and it seems to have some shape of real line having finite omissions.
    And I tried [tex] A_{n}[/tex] = R-[-1/n, 1/n], and [-1/n, 1/n], but found out all of these are not counterexample..
    Last edited: Oct 3, 2007
  2. jcsd
  3. Oct 3, 2007 #2


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    Look at random subsets of R, and study what their interiors look like.
  4. Oct 3, 2007 #3
    A(n) = R-{n}

    A(n)=R-{n} is the counterexample...
    I tried this case many times but it was not the counterexample,
    but after some fresh air, it is clear this is counter example..
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