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Thread: path connectedness

  1. #1
    Junior Member
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    path connectedness

    This is a pretty trivial question (I think?). I just want to see anyone do it with a bit of detail to determine if I am on the right track. thanks!

    Let X={a,b} and assume that X has the topology T={0, {a},X}. Show that X is path connected in this topology.
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  2. #2
    Senior Member JaneBennet's Avatar
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    Try the map $\displaystyle p:[0,1]\to X,$ $\displaystyle p(t)=a$ if $\displaystyle t\ne1$ and $\displaystyle p(1)=b.$

    Then $\displaystyle p^{-1}(\O)=\O,$ $\displaystyle p^{-1}(\{a\})=[0,\,1),$ $\displaystyle p^{-1}(X)=[0,\,1]$ which are all open in $\displaystyle [0,\,1].$
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