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

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.

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).
• @yoyo, yes make it a countable connected Hausdorff space. They also exist, see $\pi$-base – Henno Brandsma Jun 10 at 11:31