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).  
