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Math Help - Open sets and closed sets

  1. #1
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    Open sets and closed sets

    Let f: R -> R be defined by f(t)=t^2. Let U be any non empty open subset of R. Then which of the following options is correct?
    a) f(U) is open
    b) f^(-1) (U) is open
    c)f(U) is closed
    d) f^(-1)(U) is closed.

    Hey
    f(t) is a continuous function from R -> R. This implies for any open subset U in R f^(-1) (U) is open.
    So i think b) is correct. Am I right? Or is f(U) open?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by poorna View Post
    Let f: R -> R be defined by f(t)=t^2. Let U be any non empty open subset of R. Then which of the following options is correct?
    a) f(U) is open
    b) f^(-1) (U) is open
    c)f(U) is closed
    d) f^(-1)(U) is closed.

    Hey
    f(t) is a continuous function from R -> R. This implies for any open subset U in R f^(-1) (U) is open.
    So i think b) is correct. Am I right? Or is f(U) open?
    Yes, a function is continuous in the "usual analysis" sense if and only if the preimage of every open set is oepn. So, b) is correct as you said.

    For a) I would say "not a chance"

    Spoiler:


    f\left((-1,1)\right)=?



    For c) I'd like to hear your opinion

    For d) here's some food for thought.

    Spoiler:


    If U\subseteq\mathbb{R} is closed then \mathbb{R}-U is open. But, this means that f^{-1}\left(\mathbb{R}-U\right) is open. But, how can we simplify this last expression?

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