Do you have an example of a path-connected non-hausdorff space on which two points can't be injectively path-connected? (that is, any path between them is not injective). I tried to figure out what such a space should look like, and what its topological properties should be, but I failed.
Thank you very much, AF
On
Perhaps the most natural example is the interval with a doubled endpoint. It is a (non-Hausdorff) path-connected compact smooth manifold, so it basically has all the nice properties you can imagine a space which is path-connected but not arc-connected can have, but it is not hard to see that there is no arc connecting the twin endpoints.
A more pathological (but still not trivial, as in FShrike's answer) example is the Sierpiński two-point space, i.e. $X=\{x,y\}$ where the open sets are $\emptyset, X, \{x\}$. Then the function $\gamma\colon [0,1]\to X$ given by $\gamma(1)=y$, $\gamma(t)=x$ for $t<1$ is a path, but clearly there is no injective path, since $X$ is finite.
Sure. It's not a very interesting example, but here goes: take the indiscrete (a.k.a trivial) topology on any countable set. It is trivially path connected, since every function to it is continuous... but there are no injections $[0,1]\to F$ where $F$ is countable.