Is there for every infinite cardinal $\kappa$ a connected Hausdorff space $(X,\tau)$ with $|X| = \kappa$ and a collection ${\cal D}$ of mutually disjoint open sets with $|{\cal D}| = \kappa$?



Connected Hausdorff spaces with large collection of disjoint open sets
Based on the [previous answer](https://mathoverflow.net/a/358533/129074) of Wlod AA, such a space always exists for $\kappa$ infinite (it is clear that it cannot exist for $2\leq\kappa<\omega$).


Let $I$ be a countable Hausdorff connected space (see [A connected countable Hausdorff space](https://www.ams.org/journals/proc/1953-004-03/S0002-9939-1953-0060806-9/S0002-9939-1953-0060806-9.pdf), R. H. Bing, 1952), and $x$ any point in $I$. Then the quotient $(I\times\kappa)/(\lbrace x\rbrace\times\kappa)$ is such a space ($\kappa$ is considered as a discrete space), with $\mathcal D$ the collection of open sets $(I\setminus\lbrace x\rbrace)\times\lbrace\alpha\rbrace$ for $\alpha<\kappa$.


