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Can we say the set of natural numbers is homeomorphic to the set of integers? A map $f$ from $N$ to $Z$ defined $f(n)=-n$ does not work. Could you give me any hint?

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    Are you looking for a bijection or a homeomorphism? If the latter, you need to specify a topology of sorts...2012-10-29

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HINT: Any bijection works, since both sets have the discrete topology. Try matching up the even natural numbers with the non-negative integers and the odd natural numbers with the negative integers, for instance.

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    @BrianM.Scott does this require the set N to contain {0} ?2017-06-10
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Hint: Natural numbers can be even or odd. Integers can be positive or negative.

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    sorry because of the misunterstanding. thanks, now its clear.2012-10-29
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Try $\phi(n) = (-1)^n \lfloor \frac{n}{2} \rfloor$. Then $\phi:\mathbb{N} \to \mathbb{Z}$ is a bijection.