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Math Help - separation axioms

  1. #1
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    separation axioms

    GIVE AN EXAMPLE :
    X:normal
    Y:normal
    BUT X X Y : not normal
    normal is not productive
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  2. #2
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    Quote Originally Posted by flower3 View Post
    GIVE AN EXAMPLE :
    X:normal
    Y:normal
    BUT X X Y : not normal
    normal is not productive
    The classic example (given in Kelley's General Topology, and attributed by him to Dieudonné and Morse, independently) uses ordinal spaces. Let \Omega_0 be the set of all ordinal numbers less than the first uncountable ordinal \Omega and let \Omega' = \Omega_0\cup\{\Omega\}, each with the order topology. Then \Omega_0 and \Omega' are both normal, but \Omega_0\times\Omega' is not. (Let A = \{(x,x):x\in\Omega_0\} and B = \Omega_0\times\{\Omega\}. Then A and B are closed and disjoint in \Omega_0\times\Omega' but do not have disjoint open neighbourhoods. The proof is not that easy. You would do best to look it up in Kelley's book.)
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  3. #3
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    Quote Originally Posted by flower3 View Post
    GIVE AN EXAMPLE :
    X:normal
    Y:normal
    BUT X X Y : not normal
    normal is not productive
    Another example is a Sorgenfrey plane \mathbb{R}_l^{2}.

    Even though a Sorgenfrey line is normal, the Sorgenfrey plane is not normal.
    Let \Delta = \{(x, -x) | x \in R \}. As shown by the above link, no disjoint open sets can separate the disjoint closed sets K= \{(x, -x) | x \in \mathbb{Q} \} and \Delta \setminus K in \mathbb{R}_l^{2}. Intuitively, unions of the open rectangles containing K and unions of the open rectangles containing \Delta \setminus K in the above link somehow overlaps all the time. A rigorous proof needs more elaboration.

    In contrast, \mathbb{Q} and \mathbb{R} \setminus \mathbb{Q} are disjoint closed sets in a Sorgenfrey line \mathbb{R}_l, which can be separated by disjoint open sets in \mathbb{R}_l (they are both clopen sets).
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  4. #4
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    separation axioms

    thank you>>>
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