My question is the following:



> 
> Does there exist a connected metric space $\ X,\ $ where $\ |X|>1,\ $ which contains no separable connected subspace $\ Y\ $ with $\ |Y|>1\ $?
> 
> 
> 



Connected metric spaces without connected separable subspaces
The answer is *yes*, and an example can be found in 


Simon, Petr: [A connected, not separably connected metric space](http://www.openstarts.units.it/dspace/handle/10077/4292),
*Rend. Istit. Mat. Univ. Trieste* **32** (2001), suppl. 2, 127–133. 


Quoting from the introduction:



> 
> A separably connected space is a topological space, where every two points may be joined by a separable connected subspace. We present an example of a connected, but not separably connected metric space and of a connected metric space, which contains no connected separable subspaces other than one-point ones.
> 
> 
> 


