Lemma 1

Let X be a connected topological space and $\Omega$ a dense open subset of X. If, for any $x\in X$, there exists a neighbourhood V of $x$ such that $V\cap \Omega$ is connected, then $\Omega$ is connected.

Indeed, let $\Omega_0$ be a non-empty open and closed subset of $\Omega$. Let $x\in X$ and let V be a neighbourhood of $x$ such that $V\cap \Omega$ is connected. If $x\in \overline{\Omega}_0$,

so $V\cap \Omega \subset \Omega_0$. Thus, since $\Omega$ is dense in $X,\overline{\Omega}_0$ is a neighbourhood of $x$. Consequently, $\Omega_0$ is non-empty, open and closed, and since X is connected, $\overline{\Omega}_0= X$. Since $\Omega_0$ is closed in $\Omega$, this implies that $\Omega_0=\Omega \cap \overline{\Omega}_0=\Omega$, which proves that $\Omega$ is connected.