Is there a topology $\tau$ on $\omega$ such that $(\omega,\tau)$ is Hausdorff and path-connected?



Countable path-connected Hausdorff space
No, a path-connected Hausdorff space is arc-connected, whence it would be of (at least) continuum cardinality provided it has more than one point. This follows from a more general (and deep) result that a Peano space (a compact, connected, locally connected, and metrizable space) is arc-connected if it is path-connected, together with the observation that a Hausdorff space that is the continuous image of the unit interval is a Peano space. See this [section](http://ncatlab.org/nlab/show/connected+space#arcconnectedness) of the nLab, and references therein. 


