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Math Help - The zariski topology

  1. #1
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    The zariski topology

    Hi guys,

    I'm having trouble trying to prove this:

    Show that the Zariski Topology on \mathbb{A}^2 is not the product topology on \mathbb{A}^1 \times \mathbb{A}^1.

    I think my main problem is that I have no idea how you would apply the product topology on \mathbb{A}^1. Stupidly, the university have scheduled Algebraic Geometry and Topology such that they run concurrently, which means we've just been introduced to the concept of a Topological space. A push in the right direction would be much appreciated, and a brief explanation of how I could apply the concept of a product topology on affine algebraic varieties would be really good.

    Thanks a lot in advance.
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  2. #2
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    OK, I've taken the hint on the question sheet, and this is how far I've got:

    Let us pick the diagonal \mathbb{V}(x-y), given by the set \mathbb{V}_{\triangle} = \{ (x,y) : x-y=0 \}. Following the definition of the Zariski topology on \mathbb{A}^2, this is closed. We just have to show that \mathbb{V}_{\triangle} is not closed on the product topology \mathbb{A}^1 \times \mathbb{A}^1.

    Here is where I'm stuck. Where do I go from here?

    Thanks in advance,
    HTale.
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  3. #3
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    I just solved a similar question, and although this post is well over two years old (!) there is a fully worked solution here.
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