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In connected and locally connected toposes are path connected, the authors cite two spaces which are connected but not path connected - infinite sets with codiscrete topologies, and the extended long line. A space in the first class can never be sober, and topoi generalize sober spaces, so let's ignore that. The introduction to the paper, however, says the essential reason for the validity of the title as a proposition is that there are no cardinality problems in the topos setting. The extended long line isn't path connected because the last interval is too far from everything else in a strictly set-theoretic sense - the image of a path would have greater cardinality than it's domain, contradicting surjectivity.

This answer presents a bunch of connected but not path-connected spaces. I'm no topologist and to be honest I don't really care about all these pathologies, at least not enough to study each of them. What I would like is to understand to what extend "cardinality problems" are the only problems for a connected sober space to be path connected, and for a connected and locally connected space to be path connected.

  1. This $\pi$-base search gives a bunch of examples, so local connectedness really is important. Why? How does it get us "closer" to path connectedness?
  2. This $\pi$-base search only gives the extended long line and the lexicographical order on the unit square as counterexamples. The latter doesn't seem very geometric, but the issue doesn't seem to be a cardinality problem. What's going on here? Are these two examples in some sense the only ones?
Steven Clontz
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Arrow
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  • The problem with the long line isn't cardinality. At least, with choice it's not cardinality, don't know what might happen without choice. The long line has cardinality $\aleph_1 \cdot 2^{\aleph_0} = 2^{\aleph_0}$, like $[0,1]$. – Daniel Fischer Sep 04 '16 at 14:19
  • Isn't the closure of ${,(\tfrac1t,\sin t)\mid t>0}$ in $\Bbb R^2$ the standard example of a connected Hausdorff space that is not path-connected? – Hagen von Eitzen Sep 04 '16 at 14:23
  • @HagenvonEitzen I think this space is not locally connected. – Arrow Sep 04 '16 at 14:31
  • @DanielFischer I think by cardinality issues the authors meant what I wrote - a problematic path cannot exist not because of continuity issues, but rather because it's a function of sets, and so its image cannot have strictly greater cardinality than its domain. – Arrow Sep 04 '16 at 14:32
  • But the cardinality of the long line is equal to the cardinality of the unit interval. The issue is that every countable subset of $\omega_1$ is bounded. – Daniel Fischer Sep 04 '16 at 14:37
  • @DanielFischer sorry for dismissing your remark and for the silly mistake! I would still like to understand whether the two linked examples are in some sense the only ones, and why local connectedness gets us closer to path connectedness. – Arrow Sep 04 '16 at 15:14

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