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Some friends and I discovered this question when we were studying for an exam and were trying to find examples for all combinations of topological properties we had seen in the course so far. One we couldn't answer was, if there is a space that has the three properties:
-it is locally connected
-it is pathwise connected
-it is NOT locally pathwise connected
Even after looking at several books and asking our professor, we could neither find an example nor prove that such a space cannot exist.
Does anybody know something about this?

Edit: "Counterexamples in Topology" was among the books, but while I spent some time searching through it, I cannot exclude the possibility to have missed just the right space.

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    $\begingroup$ Was "Counterexamples in topology" among the sevreral boos? $\endgroup$ Commented Dec 1, 2013 at 22:08
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    $\begingroup$ Next time, you could use austinmohr.com/home/?page_id=146 It's a searchable database of the counterexamples that you can find in the book. Unafortunately, this time, your counterexample was not known... $\endgroup$ Commented Dec 2, 2013 at 0:23

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The cone on $\mathbb{R}_{cocountable}$ should work.

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