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Math Help - cardinality of continuous functions f:Z->Z

  1. #1
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    cardinality of continuous functions f:Z->Z

    What is the cardinality of the set of all functions from Z to Z

    Dont really know how to start.
    Any hints would be appreciated

    Thanks in advance.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by firebio View Post
    What is the cardinality of the set of all functions from Z to Z

    Dont really know how to start.
    Any hints would be appreciated

    Thanks in advance.
    Well, assuming that \mathbb{Z} has the normal metric (topology) then it is discrete and so every function is continuous and so the answer is \aleph_0^{\aleph_0}=2^{\aleph_0\aleph_0}=\mathfrak  {c}
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    Well, assuming that \mathbb{Z} has the normal metric (topology) then it is discrete and so every function is continuous and so the answer is \aleph_0^{\aleph_0}=2^{\aleph_0\aleph_0}=\mathfrak  {c}
    I am not familiar with topology, is there another way around this proof?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by firebio View Post
    I am not familiar with topology, is there another way around this proof?
    It isn't about topology. It's about the fact that if you aren't doing topology then I assume your idea of continuity is that of the usual idea with \delta-\varepsilon proofs. But, with this kind of continuity (which is what I assumed before) every function f:\mathbb{Z}\to\mathbb{Z} is continuous and so my previous answer stands.
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