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Thread: Pathwise connected proof

  1. #1
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    Pathwise connected proof

    Ok, so I would like to know how to prove the following proof. If anyone has any insight I would greatly appreciate it.

    Suppose $\displaystyle D$ is pathwise connected and $\displaystyle f\rightarrow R$ is continuous. Prove that $\displaystyle f(D)$ is an interval.

    Is it enough to create a function that is continuous and then show that the function lives on an interval?

    For example, $\displaystyle \phi (t)=(1-t)\alpha +t\beta $ where $\displaystyle \phi (0)=\alpha$and $\displaystyle \phi (1)=\beta $. Then show that the function lives on the interval [0,1]?
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  2. #2
    Senior Member roninpro's Avatar
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    You have to show that this works for any arbitrary function. Picking a particular function (as you did) will not suffice.

    You are really being asked to show two things:

    1) If $\displaystyle D$ is pathwise connected, $\displaystyle X$ is a space, and $\displaystyle f\to X$ is continuous, then $\displaystyle f(D)$ is pathwise connected.

    2) A subset of $\displaystyle \mathbb{R}$ is pathwise connected if and only if it is an interval.

    Give it a try.
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  3. #3
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    I have the proof of part 2 in my notes that I obtained from previous guidance. So I guess part one is what I struggle with. Its not hard for me to believe that it is true but I don't know how to get started. Maybe a proof by contradiction?
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  4. #4
    Senior Member roninpro's Avatar
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    Recall the definition of a pathwise connected set $\displaystyle X$: we say that $\displaystyle X$ is pathwise connected if for points $\displaystyle a,b\in X$, there exists a continuous function $\displaystyle \gamma:[0,1]\to X$ such that $\displaystyle \gamma(0)=a$ and $\displaystyle \gamma(1)=b$. Our goal will be to produce such a function for our $\displaystyle X$.

    You have two important facts: $\displaystyle D$ is pathwise connected, so given $\displaystyle a,b\in D$, there exists a continuous function $\displaystyle \gamma_D:[0,1]\to D$ such that $\displaystyle \gamma(0)=a$ and $\displaystyle \gamma(1)=b$; and $\displaystyle f\to X$ is continuous. How can you come up with $\displaystyle \gamma:[0,1]\to f(X)$ connecting two points?
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Not to butt in, but I feel as though this is simpler. We (and by we I mean you, zebra, and I) have already proven that a path connected space is connected. So, $\displaystyle D$ is connected and so by continuity $\displaystyle f(D)$ is connected. And, surely you know every connected subspace of $\displaystyle \mathbb{R}$ is an interval.
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