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Math Help - Simple Question

  1. #1
    Senior Member slevvio's Avatar
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    Simple Question

    if X is a topological space, is X \times [0,1] a topological space, where [0,1] is the unit interval?

    Would X \times [0,\frac{1}{2}] \subseteq X \times [0,1] be closed or open or both?


    I am interested because im trying to prove that homotopy is an equivalence relation (Transitivity)

    Thanks for any help
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  2. #2
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    Quote Originally Posted by slevvio View Post
    if X is a topological space, is X \times [0,1] a topological space, where [0,1] is the unit interval?

    Would X \times [0,\frac{1}{2}] \subseteq X \times [0,1] be closed or open or both?

    I am interested because im trying to prove that homotopy is an equivalence relation (Transitivity)

    Thanks for any help
    You mean X\times [0,1]...wrt what topology?!

    Tonio
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  3. #3
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    If you mean that [0, 1] has the "usual" metric topology on the real numbers, restricted to [0, 1], and that X\times [0, 1] is to be given the usual "product topology", then, yes, of course, it is a topological space. If A and B are are any two topological spaces then A\times B "with the product topology" is, by definition, a topological space. Since [0, 1/2] is not an open set in the usual topology on [0, 1], no, X\times [0, 1] is not an open set. Since [0, 1/2] is a closed set in the usual topology on [0, 1], yes, X\times [0, 1/2] is a closed set.
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  4. #4
    Senior Member slevvio's Avatar
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    I think you mean X \times [0,\frac{1}{2}] is not an open set.

    But thanks for your post this clears things up for me!!
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