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Let $\kappa >\aleph_0$ be a cardinal. Is there a connected space $(X,\tau)$ with $|X| = \kappa$ such that for every dense set $D\subseteq X$ we have $|D|=|X|$?

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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 $|\kappa|$.

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