Is there a topology $\tau$ on $\omega$ such that $(\omega,\tau)$ is Hausdorff and path-connected?
[Math] Countable path-connected Hausdorff space
gn.general-topology
gn.general-topology
Is there a topology $\tau$ on $\omega$ such that $(\omega,\tau)$ is Hausdorff and path-connected?
Best Answer
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 of the nLab, and references therein.