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Locally path connected

  1. Jul 4, 2013 #1
    I am trying to prove that any compact metric space that is also locally connected,must be locally path connected.
    can someone help?
    thank's in advance.
  2. jcsd
  3. Jul 4, 2013 #2
    Do you know that a locally connected and connected metric space is locally path connected?

    To prove that, consider an element x the set of all y such that there is a path from x to y.
  4. Jul 4, 2013 #3
    You mean to show that this is clopen?This is my difficulty.I suppose the compacity is needed
    Last edited: Jul 4, 2013
  5. Jul 5, 2013 #4
    Check Willard theorem 31.2.
  6. Jul 6, 2013 #5
    hahn mazurkiewicz theorem gives the answer
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