Let $\kappa >\aleph\_0$ be a cardinal. Is there a connected space $(X,\tau)$ with $|X| = \kappa$ such that for every [dense set](https://en.wikipedia.org/wiki/Dense_set) $D\subseteq X$ we have $|D|=|X|$?



Connected spaces where every dense set is large
Yes. Take any countable connected Hausdorff space $C$, fix any point $c\in C$, and consider the quotient space $X=C\times \kappa/ \{c\}\times \kappa$. Here the cardinal $\kappa$ is endowed with the discrete topology. 


It is easy to see that the space $X$ is connected and Hausdorff and each dense subset of $X$ has cardinality $|\kappa|$.


