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Can you give me an example of a space that is locally path connected but not path connected, if it exists ?

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Perhaps a more interesting question is are there path connected spaces that aren't locally path connected at any point? – JSchlather Jun 26 '12 at 16:12
This website is useful for questions like these. – Zev Chonoles Jun 26 '12 at 16:13
Wow! I knew about Topospaces, which has something similar. But the site from @ZevChonoles' comment looks great! – Martin Sleziak Jun 26 '12 at 18:33
up vote 3 down vote accepted

Simple examples include $\mathbb{R} \setminus \{0\}$ with its natural topology as a subspace of $\mathbb{R}$, and any set with at least two points with the discrete topology.

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It may also be worth pointing out that this is the only example in a sense: A locally path connected space is path connected iff it is connected. – Sam Jun 26 '12 at 16:13

Just for completeness, the website Zev Chonoles references (called $\pi$-Base) gives the following spaces as locally path connected but not path connected. (It should be noted that this information is also available in Steen and Seebach's Counterexamples in Topology.) You can learn more about these spaces by viewing the search result.

Countable Discrete Topology

Either-Or Topology

Finite Discrete Topology

Hjalmar Ekdal Topology

Odd-Even Topology

Sierpinski's Metric Space

Uncountable Discrete Topology

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