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Math Help - Help understanding this theorem/proof

  1. #1
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    Help understanding this theorem/proof

    Here's the theorem that I have to prove:

    A set U \subset R is called an open set if \forall x \in U \  \exists \epsilon >0 such that (x-\epsilon, x+\epsilon) \subset U. Prove that the following two statements are equivalent:
    (a) \rightarrow R" alt="f \rightarrow R" /> is continuous
    (b) For any open set U \subset R, there exists an open set W \subset R such that f^{-1}(U)=D \cap W. Here, f^{-1}(U)=\{x \in D|f(x) \in U\} denotes the preimage of U under f.

    Can anyone help me with this? I don't even understand the "idea" of it, much less how to prove it.
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  2. #2
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    Anyone able to help with this? Does this theorem have a name, by chance?
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  3. #3
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    Quote Originally Posted by paupsers View Post
    Here's the theorem that I have to prove:

    A set U \subset R is called an open set if \forall x \in U \  \exists \epsilon >0 such that (x-\epsilon, x+\epsilon) \subset U. Prove that the following two statements are equivalent:
    (a) \rightarrow R" alt="f \rightarrow R" /> is continuous
    (b) For any open set U \subset R, there exists an open set W \subset R such that f^{-1}(U)=D \cap W. Here, f^{-1}(U)=\{x \in D|f(x) \in U\} denotes the preimage of U under f.

    Can anyone help me with this? I don't even understand the "idea" of it, much less how to prove it.
    Which statement of continuity do you know? b) is a pretty standard definition of continuity from a subspace topology. I am guessing that D is a subset of R with the subspace topology.
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    bump
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