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?
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.