Given an infinite cardinal $\kappa$, is there a connected Hausdorff space $(X,\tau)$ with $|X|=\kappa$, and for every infinite cardinal $\lambda \leq \kappa$ there is an open set $U\in \tau$ with $|U| = \lambda$?



Connected Hausdorff spaces with different cardinalities of open sets
Let $C$ be Bing's [countable connected Hausdorff space](http://www.ams.org/journals/proc/1953-004-03/S0002-9939-1953-0060806-9/S0002-9939-1953-0060806-9.pdf). Specify some point $0 \in C$. Let $D(\alpha)$ be the discrete space of cardinality $\alpha$.
Let $F(\alpha)$ be the product $C \times D(\alpha)$ with the set $\{0\}\times D(\alpha)$ treated as a point $r(\alpha) \in F(\alpha)$. It is easy to see $F(\alpha)$ is connected and Hausdorff and has an open set $(C-0) \times D(\alpha)$ of cardinality $\aleph\_0 \times \alpha = \alpha$.


Now let $\displaystyle D = \bigsqcup\_{\alpha< \kappa}F(\alpha)$ be the disjoint union and treat the set $\{ r(\alpha): \alpha < \kappa)\}$ as a point. All the images of $(C-0) \times D(\alpha)$ are open in the quotient space and $D$ itself is open with cardinalty $\kappa$.


**Edit:** Here's an easier one. Start with $C \times D(\kappa)$ and treat $\{0\} \times D(\kappa)$ as a point. Then the sets $(C-0) \times \{d\}$ are open and disjoint and countable. By taking unions you get open sets of all possible cardinalities.


