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Math Help - Topology question

  1. #1
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    Topology question

    How do I show that convergence is a topological property?

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  2. #2
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    Re: Topology question

    Quote Originally Posted by hairymclairy View Post
    How do I show that convergence is a topological property?
    What are the exact properties of a homeomorphism?
    Now think:"How are continuity and convergence related?"
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  3. #3
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    Re: Topology question

    Let f:X\to Y be a homeomorphism between topological spaces X and Y. If \left(x_n\right)_{n=1}^\infty is a sequence in X converging to x\in X, show that the sequence \left(f(x_n)\right)_{n=1}^\infty in Y converges to f(x)\in Y.

    The sequence \left(x_n\right)_{n=1}^\infty converges to x\in X iff for every open set U\subseteq X containing x there exists a natural number N such that x_n\in U for all n>N.
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