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.

  • $\begingroup$ Maybe some modification of the Knaster-Kuratowski fan would work? Gerhard "Do I Feel A Breeze?" Paseman, 2017.01.06. $\endgroup$ – Gerhard Paseman Jan 6 '17 at 9:45
  • $\begingroup$ Sorry, I forgot (again) to add infinity as a requirement $\endgroup$ – Dominic van der Zypen Jan 6 '17 at 12:43

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.

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