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Cardinality/Function Question

  1. Sep 20, 2010 #1
    Function Types: continuous, surjective, injective, bijective, continuous surjective, continuous injective, continuous bijective. Then all of the above -- with each possible type of inverse?

    What is possible with F:X-->Y where X, Y can be [0,1], (0,1), [0,1), Q, R, N?

    I certainly don't expect a full listed answer for each combination, but some general principles would be great. :)

    I already know there couldn't be bijections between sets of different cardinality. And I know there couldn't be an injection from greater to lower, or a surjection from smaller to greater cardinality.


  2. jcsd
  3. Sep 21, 2010 #2
    Specifying, for instance [0,1), is not enough. You need to specify the topology. The same set can be equipped with different topologies.
  4. Sep 21, 2010 #3
    I'm assuming he means the standard Euclidean topology
  5. Sep 21, 2010 #4
    Well, then maps can be classified, for instance, by their differentiability properties - whenever applicable, and there are infinitely many classes. But why would one ask such question?
  6. Sep 21, 2010 #5
    Yes the Euclidean metric. I'm not really interested in differentiability types right now -- just the ones listed (surjective, injective, etc...) Maybe this is the wrong folder to ask -- but the question came to me while studying topology so it seemed appropriate.

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