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    • Thread Starter

    consider R2 (real numbers -2dimensional) with standard euclidean distance
    and a subset

    A={(x,y): x\=0, y/x belongs to Q (rational numbers) }

    i need help finding its closure and interior, i think interior will be empty set, but i'm still not sure how to show both.
    • PS Helper

    PS Helper
    Think about this geometrically: your set is essentially the set of lines through the origin which have rational gradient, but with the point at the origin removed.

    Now, pick any point in \mathbb{R}^2 and a ball of arbitrary radius about that point. Must it contain some point which lies on a line with rational gradient? So what you can you say about the closure of A?

    And also, pick any point in A. Can you find some ball around that point which doesn't contain any points which lie on lines of irrational gradient? So what you can you say about the interior of A?
    • Thread Starter

    Thank you! i understand now.
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Updated: March 20, 2011
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