**Question:** Is there a path connected subset of $\mathbb R^2$, without any bounded path connected subset ?

**Motivation:** If we replace "path connected" by "connected", then the answer is positive. It was proven by Stefan Mazurkiewicz, see [the original article][1]. 


  [1]: http://matwbn.icm.edu.pl/ksiazki/fm/fm2/fm2112.pdf