Let $X$ an infinite hyperconnected topological space and let $F$ a finite subset of $X$ such that $F$ is not dense and with empty interior (or, equivalently, $X - F$ is dense).
Can we prove that $F = \emptyset$ or show a counterexample?
As an example, let $X = \mathbb{R}$ with standard topology and $F =$ { $0$ }. So, $F$ is not dense and $X - F$ is dense. So $F$ is not empty... but $\mathbb{R}$ is not hyperconnected.
If we equip $\mathbb{R}$ with cofinite topology then we obtain an example of hyperconnected space, and in that case the previous example doesn't work because $F =$ { $0 $ } is dense (or, at least, it seems to me). But also $F = ${ $0, 1$ } is not dense, and, in general, I think every finite $F$ that is non empty should be dense.
While I'm not able to find any example of an infinite hyperconnected space with a finite subset $F$ that is both non-dense and co-dense, this doesn't mean that it can't exist... and I wonder if, in this case, $F$ necessarily should be the empty set.
EDIT: There was an error in my second example. In that case, $F =$ { $0 $ } is not dense and $X - F$ is dense and so we have an example of a non-empty, non-dense, co-dense finite subset of an hyperconnected space. Thanks to Eric Wofsey to point out my bad mistake