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Math Help - Path connected space

  1. #1
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    Path connected space

    I'm starting to learn about path connected spaces and am a bit sketchy, any help would be appreciated, cheers,

    Lets say i Have the set S = {0,1} with topology { emptyset, {0}, S }

    and i need to prove S is path connected and therefore connected,

    to show path connectedness,
    i need to show a path exists from x to y for any x,y in S,

    to show this i need a cotinuous map p : [0,1] -> S such that p(0) = x and p(1) = y
    im confused how to do this for this set,
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  2. #2
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    Re: Path connected space

    Let f(x)= 0 for 0\le x< 1/2 and f(x)= 1 for 1/2 \le x\le 1.

    Show that this function is continuous in this topology.
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  3. #3
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    Re: Path connected space

    this is the problem i'm having, do i show this is continuous using epsilon delta? or is that not appropriate here and i should be using defn of continuity where the pre-image of the open sets are open?
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