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Math Help - IVT with different topologies

  1. #1
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    IVT with different topologies

    Prove whether or not the general version of the IVT holds when the range \mathbb{R} is given each of the following topologies:

    1. The trivial topology
    2. The discrete topology
    3. The Lower limit topology
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Prove whether or not the general version of the IVT holds when the range \mathbb{R} is given each of the following topologies:

    1. The trivial topology
    2. The discrete topology
    3. The Lower limit topology
    Let f:[a,b] \rightarrow \mathbb{Re} be a continuous function that IVT holds.

    Since the image of a (path) connected space under a continuous function is (path) connected, IVT does not hold for (2) and (3) ( \mathbb{R} with a discrete and a lower limit topology are totally disconnected).

    So, I think IVT holds for 1 only.
    Last edited by aliceinwonderland; April 11th 2009 at 03:11 AM.
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