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Math Help - Order Topology vs. the standard one?

  1. #1
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    Order Topology vs. the standard one?

    on R^2 we define an order topology (lexicographic order) as follow: (x1,y1)>(x2,y2) if x1>x2 or ( x1=x2 & y1>y2).
    Is this topology equivalent to the standard topology? if not, which topology is finer?(give an example of an open set in one but not in the other)?

    thanks
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  2. #2
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    Quote Originally Posted by aharonidan View Post
    on R^2 we define an order topology (lexicographic order) as follow: (x1,y1)>(x2,y2) if x1>x2 or ( x1=x2 & y1>y2).


    You have defined no topology at all: you've defined a (partial) order on \mathbb{R}^2 .

    Now, as far as I know, an order topology is defined on a totally ordered set, so if you meant

    this then you first prove the above gives you a total order on the real plane and then look at

    the derived (order) topology, and then you can ask yourself whether this toplogy is the

    same as the usual Euclidean one.

    Tonio



    Is this topology equivalent to the standard topology? if not, which topology is finer?(give an example of an open set in one but not in the other)?

    thanks
    .
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  3. #3
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    I wasn't clear enough. if (X,<) is a partial order set, then all (a,b) a<b U (-infinity,infinity) is a base of a topology on X.
    now I need to compare this topology with the Euclidean one.
    any help or hint is appreciated
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