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What would this open set in R look like?

  1. Jun 10, 2010 #1
    Well order the real numbers, let [tex] {a_n}_{n \in S_{\Omega} } [/tex] be the the singleton sets of odd numbers in the well order (i.e. skip a number, grab a number, skip a number, grab a number). Since the real numbers are [tex] T_1 [/tex] then singleton sets are closed. [tex] \mathbb{R} [/tex] as a topological space is closed by definition. Let [tex] U := \mathbb{R} \bigcap_{n \in S_{\omega} } {a_n} [/tex] . Then U is closed and hence [tex] \mathbb{R} \backslash U [/tex] is open.

    Is my logic correct here? It just seemed strange to me that this set would be open. Does anyone know what this set would look like? Thanks.
     
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  3. Jun 10, 2010 #2

    Hurkyl

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    Your construction is somewhat confusing, and in my best guess at your intent, U is the empty set.


    Incidentally, what are you doing about infinite ordinals? I suppose you could decree limit ordinals to be even, and extend the notion of being even/odd from there.
     
  4. Jun 11, 2010 #3

    Landau

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    As a well ordening of the reals requires the axiom of choice, I don't think one could imagine how the set looks like.
     
  5. Jun 11, 2010 #4

    HallsofIvy

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    More importantly, while any set can be "well ordered", that does NOT imply that, given one number, there exist a "next" number. That is only possible for "countable" sets and the real numbers are not countable.

    You could ask this question of the rational numbers but then I think that Landau's comment is valid.
     
  6. Jun 11, 2010 #5

    Hurkyl

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    That one's guaranteed -- every ordinal has a successor. It's the reverse that can fail: limit ordinals do not have a predecessor.
     
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