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Cardinality of continuous functions

  1. Feb 13, 2010 #1
    1. The problem statement, all variables and given/known data
    What is the cardinality of the set of all continuous real valued functions [tex] [0,1] \rightarrow R [/tex].




    3. The attempt at a solution
    In words:
    I will be using the Cantor Bernstien theorem. First the above set, lets call it A, is lesser then or equal to the set of all functions from R to R which has a cardinality of [tex] \aleph ^ \aleph [/tex]
    Also, the cardinality of all continuous functions [tex] [0,1] \rightarrow R [/tex]. is lesser then or equal too the cardinality of the set of all functions, continuous and not continuous, from [tex] [0,1] \rightarrow R [/tex].
    The cardinality of the segment [0,1] is [tex] \aleph [/tex] and so the cardinality of all functions [tex] [0,1] \rightarrow R [/tex]. is also [tex] \aleph ^ \aleph [/tex] and so by the cantor bernsiten theorem the cardinality of A is also [tex] \aleph ^ \aleph [/tex]

    In formal math:
    I use C to denote continuous functions.
    [tex] \aleph ^ \aleph =|R^R| \geq |R^{C[0,1]}| = |C[0,1] \rightarrow R| = |R^{C[0,1]}| \leq |R^{[0,1]}|=\ \aleph ^ \aleph [/tex]


    Is this correct? and formal anough?
    Thanks
    Tal
     
  2. jcsd
  3. Feb 13, 2010 #2
    Ahhh I see my mistake allready.
    Is it enough to say that a coninuous function is determined by its values on Q. Therefore
    [tex] C[0,1] \rightarrow R = R^{[0,1] \in Q}= \aleph [/tex]
     
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