Can you give me an example of a space that is locally path connected but not path connected, if it exists ?
Locally Path Connected space that is not Path Connected.
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general-topology
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1Wow! I knew $a$bout [Topospaces](http://topospaces.subwiki.org/wiki/), which has something similar. But the site from @ZevChonoles' comment looks great! – 2012-06-26
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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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0It $m$ay also be worth pointing out that this is the only exa$m$ple in a sense: A locally path connected space is path connected iff it is connected. – 2012-06-26
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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