If $(X,\tau)$ is a topological space, we say $S\subseteq X$ is discrete, if the subspace topology on $S$ inherited from $(X,\tau)$ is discrete. 


Let $\kappa$ be an infinite cardinal. Is there a connected $T\_2$-space $(X,\tau)$ and a discrete subset $S\subseteq X$ such that $|X| = |S| = \kappa$?



Large discrete subsets of connected $T_2$-spaces
For any cardinal $\kappa$ at least the size of the continuum, the "really long line" of length $\kappa$ is an example.


This space, let's call it $L\_\kappa$, is defined as follows. Begin with the ordinal $\kappa$ with its usual order topology. Then, for any $\alpha \in \kappa$, connect $\alpha$ and $\alpha+1$ with a copy of the unit interval.


$L\_\kappa$ obviously has cardinality $\kappa$, and a discrete subspace of size $\kappa$ is given by the set of all successor ordinals.


If $\kappa < \mathfrak{c}$, then you can get an example by modifying the construction of $L\_\kappa$: simply replace the unit interval with a countable connected Hausdorff space. (It's not obvious, but [countable connected Hausdorff spaces do exist](http://www.ams.org/journals/proc/1953-004-03/S0002-9939-1953-0060806-9/S0002-9939-1953-0060806-9.pdf).)


