A space where no paths are homotopic I was just thinking about algebraic topology and was wondering: It is possible to find a Hausdorff space where no paths are homotopic?
Excluding trivial examples.

The answer is no. I presume by excluding trivial examples you are asking for a space $X$ with two distinct points $x,y$ such that there is a path $\gamma$ from $x$ to $y$ (note that $x,y$ being distinct is not so important, if there is a nonconstant loop at $x$ automatically this is true by just picking $y$ to be the midpoint). Given any such space, there is a homotopy of paths from the constant loop at $x$ to $\gamma^{-1} \circ \gamma$. Clearly these paths are unequal, since their images are different.
