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Fundamental groups and arcwise connected spaces.

  1. Mar 24, 2014 #1
    If a space X is arcwise connected, then for any two points p and q in X the fundamental groups ##\pi_1(X,p)## and ##\pi_1(X,q)## are isomorphic. This means that we can, up to isomorphisms, identify both groups with their equivalence class ##\pi_1(X)##.

    I started to think about the generality of this theorem. Is there any interesting/important spaces which are not arcwise connected except for the union of two disjoint spaces etc?
     
  2. jcsd
  3. Mar 24, 2014 #2
    Any Hausdorff space that is not path-connected is also not arc-connected. Therefore, the so-called "Topologist's Sine Curve," a relatively important topological space, is not arc-connected.
     
  4. Mar 24, 2014 #3

    micromass

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    It's not a relatively important topological space, since it serves more as a counterexample than as something worth studying on its own. Sure, it's a nice and important counterexample, but that's it. It's not a topological space central to algebraic topology like topological manifolds.
     
  5. Mar 25, 2014 #4
    Meh. It was the first Hausdorff space that isn't path-connected that came to mind. :rolleyes:

    I have to agree with micro, though. It isn't important itself. Come to think of it, off the top of my head, I can't think of any particularly "important" spaces that are not arc-connected.

    As a side note, topological manifolds are locally arc-connected (since they are Hausdorff and locally path-connected).
     
  6. Mar 27, 2014 #5

    mathwonk

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    what is the definition of "important"? what about the rational numbers? some people think they are important. we need more details to answer this, i think.
     
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