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Math Help - Interior, closure, open set in topological space

  1. #1
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    Interior, closure, open set in topological space

    T=\mathbb R^2 \: A=\mathbb Q^+ \times \mathbb R^+

    \overline{A}=? /closure of A/
    Int A=?
    In (A, \tau_A) is A open? (A, \tau_A) /topological space/

    Thank you in advance!
    Last edited by zadir; March 2nd 2011 at 08:22 AM.
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  2. #2
    Super Member girdav's Avatar
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    You can show that if you have two topological spaces (X,\mathcal T_1) and (Y,\mathcal T_2) and if you consider the product space (X\times Y,\mathcal T\otimes \mathcal U) you will have for any A\subset X and B\subset Y that \overline{A\times B}=\overline{A}\times \overline B and \mathrm{Int}(A\times B)=\mathrm{Int}(A)\times \mathrm{Int}(B).
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  3. #3
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    Quote Originally Posted by girdav View Post
    You can show that if you have two topological spaces (X,\mathcal T_1) and (Y,\mathcal T_2) and if you consider the product space (X\times Y,\mathcal T\otimes \mathcal U) you will have for any A\subset X and B\subset Y that \overline{A\times B}=\overline{A}\times \overline B and \mathrm{Int}(A\times B)=\mathrm{Int}(A)\times \mathrm{Int}(B).
    Thank you for your help.
    To tell the truth I don't know how to continue, how to use that in the problem.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zadir View Post
    Thank you for your help.
    To tell the truth I don't know how to continue, how to use that in the problem.
    He's saying that if you consider giving \mathbb{R}^2 the usual topology induced by the usual norm then the induced topology is the same as if \mathbb{R}^2 the product topology (when both copies of the reals are endowed with the usual topology). That said, for the product topology it's easy to prove things like \text{cl}_{\mathbb{R}\times\mathbb{R}}\left(A\time  s B\right)=\text{cl}_{\mathbb{R}}(A)\times\text{cl}_  {\mathbb{R}}(B). So now since \mathbb{Q},\mathbb{R} are both dense in \mathbb{R} you can conclude that \mathbb{Q}\times\mathbb{R} is dense in \mathbb{R}^2. etc.
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