Is there a connected $T\_2$-space $(X,\tau)$ with more than 1 point and with the following property?



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> Whenever $D\subseteq X$ is dense, $X\setminus D$ is not dense.
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Dense and co-dense subsets in connected $T_2$-spaces
A topological space is called *irresolvable* if it is not the disjoint union of two dense subsets. So you are asking whether there is a connected, $T\_2$, irresolvable space with more than one point. The answer is yes! You can find a proof, along with a few references to relevant literature, in 



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> D. Anderson, "On connected irresolvable Hausdorff spaces," *Proceedings of the AMS*, 1965. [link](https://www.jstor.org/stable/2034674?seq=1#page_scan_tab_contents)
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