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Density of a set (proof check)

  1. Oct 29, 2009 #1
    1. The problem statement, all variables and given/known data

    Given that O is an open set in W, and A is dense in W, prove that [tex]A \cap O[/tex] is dense in [tex]O[/tex]

    2. Relevant equations



    3. The attempt at a solution

    Looking at what is given,
    [tex]O \subseteq W[/tex] because O is an open set in W, and [tex]W\subseteq\overline{A}[/tex] because A is dense in W

    Suppose an arbitrary [tex]x \in O[/tex] (so [tex]x \in \overline{O}[/tex])
    So [tex]x \in W[/tex]
    Which means [tex]x \in \overline{A}[/tex]

    Thus [tex]x \in \overline{A} \cap \overline{O}[/tex] or more specifically, [tex]x \in \overline{A \cap O}[/tex]


    That proves that [tex]O \subseteq \overline{A \cap O}[/tex], which should be enough to show that [tex]A \cap O[/tex] is dense in [tex]O[/tex]

    Does that sound about right?
     
  2. jcsd
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