1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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?
  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]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook