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Math Help - Homeomorphism

  1. #1
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    Homeomorphism

    How can I prove that every subset of R^n is homeomorphic to a closed subset of R^(n+1)? Does this only works for real number only as R is metrizable?

    Thanks.
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  2. #2
    Senior Member roninpro's Avatar
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    Hello.

    I haven't thought about the problem too deeply or anything, but it seems that the map f:\mathbb{R}^n\to \mathbb{R}^{n+1} defined by f(x_1,\ldots,x_n)=(x_1,\ldots,x_n,0) would do the trick.

    Did you try it?
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  3. #3
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    But the inverse of f may not be continuous unless we restrict the range of inverse of f to the set in R^(n+1).
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  4. #4
    Senior Member roninpro's Avatar
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    Isn't that what we're doing? We want a homeomorphism of \mathbb{R}^n into some subset of \mathbb{R}^{n+1}. In this case, we implicitly chose the subset \{(x_1,\ldots,x_n,0)\in \mathbb{R}^{n+1}\ |\ x_1,\ldots,x_n\in \mathbb{R}\}.
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