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A topological space that is neither discrete nor indiscrete

  1. Dec 8, 2013 #1
    1. The problem statement, all variables and given/known data

    is it possible to have a topological space that is neither the indiscrete nor the discrete, and very set in the topology is clopen?

    2. Relevant equations



    3. The attempt at a solution

    let ##X## = {(0,1),(2,3)} with the ordinary topology on R.
    (0,1) is open, but it's complement which is (2,3) is open and which means (0,1) is closed. This (0,1) is clopen. Same argument for (2,3). Is this right?
     
    Last edited: Dec 8, 2013
  2. jcsd
  3. Dec 8, 2013 #2

    Dick

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    Science Advisor
    Homework Helper

    Yes, it is. But you aren't stating it very well. Your space is ##X##={0,1,2,3}. Your open sets are ∅,{0,1},{2,3},X. Then every open set is also closed, as you say. And that is NOT the ordinary topology on R.
     
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