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.

The density of U-- A follows from Differentiable and Analytic Manifolds, Results, 3.2.5. Assume first that $n= 1$. If $a\in A$, the power series expansion of $f$ about $a($Differentiable and Analytic Manifolds, Results, 3.2.1) is not reduced to 0, and it follows that there exists a neighbourhood $V_a$ of $a$ in U such that $f$ does not vanish on $V_a$ -- $\{a\}$. Thus, $a$ is isolated in A, which proves that A is a discrete subset of U, hence countable since U is countable at infinity. Let $x, y\in U$ -- A. The union of the real affine lines joining $x$ (resp. $y$) to a point of A is meagre (General Topology, Chap. IX, §5, no. 2, Def. 2). Hence, there exists $z\in U$ -- A such that neither of the segments $[x, z]$ and $[y, z]$ meets A. The points $x, y, z$ thus belong to the same connected component of U -- A, which proves the lemma in the case $n= 1$. We turn to the general case. We can assume that A is the set of zeros of $f($General Topology, Chap. I, §11, no. 1, Prop. 1). Let $x, y\in U$-- A and let L be an affine line containing $x$ and $y$. The restriction of $f$ to $L\cap U$ is not identically zero since $x\in L\cap U$. By what has already been proved, $x$ and $y$ belong to the same connected component of $(L\cap U)$ -- $(L\cap A)$ and hence to the same connected component of U -- A.

For any $x\in X$, there exists an analytic function germ $f_x$, not vanishing

at $x$, such that the germ of A at $x$ is contained in the germ at $x$ of the set

of zeros of $f_x$.

Then X -- A is dense in X and connected.

The density of X-- A follows from Differentiable and Analytic Manifolds, Results, 3.2.5. We can assume that A is closed (General Topology, Chap. I, §11, no. 1, Prop. 1). For any $x\in X$, there exists an open neighbourhood V of $x$ and an isomorphism $c$ from V to an open ball in $\mathbf{C}^n$ such that $c(A\cap V)$ is contained in the set of zeros of a holomorphic function not identically zero on $c(V)$. Then, by Lemma $2, V\cap (X$ -- A) is connected. In view of Lemma 1, this proves that X -- A is connected.