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Is it possible that there is a connected topological space without path-connected subspace?

Furtherly ~ Is that any connected topological space $X$ always has dense path-connected subspace?

Or

Is that any non-trivial connected topological space $X$ has the property: $\forall x\in X$ there exists a path $p:[0, 1]\to X$ such that $x$ is an accumulation point of the image of $p$ in X.

I'm wondering that one of statement above is a sufficient and necessary condition of a connected space. But I can't find any suitable keyword to google this. Can anyone advance?

If there's a counterexample, then it would be really interesting.

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Consider $X = \Bbb N$ in the cofinite topology, then $X$ is connected but any continuous $p: [0,1] \to X$ is constant. As noted in the comments, Hausdorff examples also exist, but we cannot get regular Hausdorff countable connected spaces (as then maps onto $[0,1]$ exist by normality).

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    $\begingroup$ The last fact is a consequence of Sierpinski's classic theorem that a continuum cannot be a non-trivial at most countable union of pairwise disjoint closed subsets. $\endgroup$ – Henno Brandsma Jun 10 at 11:02
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    $\begingroup$ See this thread for more info. $\endgroup$ – Henno Brandsma Jun 10 at 11:17
  • $\begingroup$ Wow~ nice example. What about add the Hausdorff condition, is it still possible to find an counterexample? $\endgroup$ – yoyo Jun 10 at 11:25
  • $\begingroup$ Thank for your reference information~ $\endgroup$ – yoyo Jun 10 at 11:28
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    $\begingroup$ @yoyo, yes make it a countable connected Hausdorff space. They also exist, see $\pi$-base $\endgroup$ – Henno Brandsma Jun 10 at 11:31

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