Are there examples of non-empty connected $G_\delta$ sets in compact Hausdorff spaces that are neither closed or open and have an empty interior?
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Sure. Any dense $G_\delta$ with empty interior qualifies. The construction of such sets in intervals of the real line isn't hard. – Daniel Fischer May 23 '14 at 10:45
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Your answeres made me realize that i forgot another requirement, so i edited my question. – Penguin May 23 '14 at 12:13
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How about a half open/half closed line segment in the unit disc? – David Mitra May 23 '14 at 12:58
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Let $X = [0,1] \times [0,1]$, and consider $$A = X \setminus ( \mathbb{Q} \times \mathbb{Q}) = \{ \langle x , y \rangle \in X : x \notin \mathbb{Q}\text{ or }y \notin \mathbb{Q} \}.$$
- As $X \cap ( \mathbb{Q} \times \mathbb{Q} )$ is countable (hence F$_\sigma$), it follows that $A$ is G$_\delta$ and is clearly nonempty.
- It is both dense and co-dense in $X$, and so is neither open nor closed.
- Since it is co-dense, it also follows that $A$ has empty interior.
- Using the ideas from JDH's answer, one can show that it is (path-)connected.
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