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"Find an example of a path connected, locally path-connected space which does not have a simply connected covering space". I was reading hatcher and he gives an example of shrinking wedge of circles, and in exercise 5 he mentioned the comb space ,both being spaces with no simply connected covering space, but these two spaces are not locally path connected, so can anyone help me in this ?

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    $\begingroup$ The Hawaiian earring (shrinking wedge of circles) is locally path-connected. $\endgroup$ – user98602 Apr 12 '15 at 21:07
  • $\begingroup$ On the other hand, the Hawaiian earring is not semi-locally simply connected. A space must have this property in order to have a universal cover. $\endgroup$ – Ayman Hourieh Apr 12 '15 at 21:21
  • $\begingroup$ I said it doesn't have a simply connected covering space therefore "not semi-locally simply connected" so the hawaiin earrings are the appropriate example,I didn't knew it is locally path connected $\endgroup$ – Butterfly Apr 12 '15 at 21:33
  • $\begingroup$ @MikeMiller i can't see how it's locally path connected it looks like the comb space to me ,can you tell me how to see it ? Thank you $\endgroup$ – Butterfly Apr 13 '15 at 20:26
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I would suggest you to go and have a look at my answer for this following question in math overflow.... link : https://mathoverflow.net/questions/111310/universal-covering-space-for-non-semilocally-simply-connected-spaces/173250#173250

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