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. 


Is there an infinite connected $T\_2$-space $(X,\tau)$ and a discrete subset $S\subseteq X$ such that no proper superset of $S$ is discrete?


**EDIT**: Added "infinite" in the question.



Maximal discrete subsets of connected $T_2$-spaces
No. If $S$ is a maximal discrete subset of a $T\_1$-space $X$, then every point of $S$ is isolated in $X$ (in fact, $S$ must be the set of isolated points of $X$ and it must be dense in $X$). Thus if $X$ has at least two points, it is not connected. 


