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Math Help - Classifying open and closed sets

  1. #1
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    Classifying open and closed sets

    Hi;

    let X={1,2,3}. The topology of X is the induced topology from the usual topology on the real line. Consider the product space X * [a,b], where a,b are real numbers. My question is classify all open sets and closed sets of X*[a,b]. what about 1*[a,b], is it open set?

    thank you in advance
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  2. #2
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    Re: Classifying open and closed sets

    If A is an open subset of X and B is an open subset of Y then A\times B is an open subset of X\times Y.

    However, any union of such products, even if it isn't such a product, is an open set.
    Last edited by HallsofIvy; March 14th 2014 at 12:30 PM.
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  3. #3
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    Re: Classifying open and closed sets

    Hint: show every subset of X is open. Why? Consider the set:

    (k-ε,k+ε)∩X where 0 < ε < 1 and k = 1,2, or 3, which is clearly open under the induced (subset) topology.

    Show this means that if S⊆X, and U⊆[a,b], that SxU is open iff U is.
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