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Math Help - Homeomorphic finer topology

  1. #1
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    Homeomorphic finer topology

    Find two different topologies on \mathbb{R} where one is strictly finer, yet the two are homeomorphic
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Find two different topologies on \mathbb{R} where one is strictly finer, yet the two are homeomorphic
    (Z, T_1) and (Q, T_2) as subspaces of R with a discrete topology.


    Let B_1 and B_2 be bases for topologies T_1 and T_2.

    T_2 is strictly finer than T_1, since for each x in Z and for each basis element \{x\} \in B_1 containing x, we have a basis element \{x\} in B_2 that contains x. The converse is not necessarily true.

    Since there is a bijection between Z and Q, we have a homeomorphism between (Z, T_1) and (Q, T_2) as subspaces of R with a discrete topology.
    Last edited by aliceinwonderland; February 24th 2009 at 10:43 PM.
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