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Math Help - Compact Sets

  1. #1
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    Compact Sets

    I am confused about how to approach the following problem:

    Assume  S \subset R^n is a compact set, and V \subset R^m is compact.
    Prove that S \times V is compact in R^{n+m}.

    First of all, what does it mean by S * V? I do not understand this notation. I have seen this notation used for defining regions like [a, b]  \times [c, d] defines region for integration or something. But not in the context of sets. How would I attempt this problem?
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  2. #2
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    Quote Originally Posted by Rozaline View Post
    I am confused about how to approach the following problem:

    Assume  S \subset R^n is a compact set, and V \subset R^m is compact.
    Prove that S \times V is compact in R^{n+m}.

    First of all, what does it mean by S * V? I do not understand this notation. I have seen this notation used for defining regions like [a, b]  \times [c, d] defines region for integration or something. But not in the context of sets. How would I attempt this problem?
    First, it is NOT "S*V", it is " S\times V" as you give in the problem itself. I don't know of any meaning for "S*V" for sets but S\times V is the "Cartesian product", the set of ordered pairs, (a, b), such that the first member, a, is from S and the the second member, b, is from V.

    ([a, b] and [c, d] are sets. They are the sets, in the number line, \{x| a\le x\le b\} and \{y|c\le y\le d\}. and [a, b]\times [c, d] is the set in the plane \{(x, y)|a\le x\le b, c\le y\le d\}.)

    And you will need information about the "product topology" to answer this question! A set of points, {(x,y)}, in A\times B is open if and only if the set of all "x" in the pairs is open in A and the set of "y" in the pairs is open in B.
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  3. #3
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    Quote Originally Posted by Rozaline View Post
    I am confused about how to approach the following problem:

    Assume  S \subset R^n is a compact set, and V \subset R^m is compact.
    Prove that S \times V is compact in R^{n+m}.

    First of all, what does it mean by S * V? I do not understand this notation. I have seen this notation used for defining regions like [a, b]  \times [c, d] defines region for integration or something. But not in the context of sets. How would I attempt this problem?

    You don't know what cartesian product of sets S x V is?? but you must master this before you attempt to study topology, otherwise it is gonna be a very long course for you.

    S x V = { (s,v) ; s in S, v in V }

    Tonio
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