An irreducible plane curve gives a connected punctured covering

Choose a disc $V$ and radius $r>0$ as in [F2], and shrink $V$ to $\{\mathrm{\mid }x\mathrm{\mid }<\epsilon \}$ so that $D_W(x)\mathrm{\ne }0$ on its punctured part $D^{\ast }$, using [F3] and [F5]. By [F4] each slice over $D^{\ast }$ has exactly $m$ distinct simple roots, all in $D$ by [F2]. At any base point [F6] supplies a holomorphic graph through each root. Intersect the finitely many base neighbourhoods and shrink until these graphs stay in $D$ and are pairwise disjoint. They exhaust each fibre, since a degree-$m$ polynomial has at most $m$ roots by [F4]. Thus they give an evenly covered neighbourhood with $m$ holomorphic sheets. This proves directly, in the fixed coordinates, that $Z(W)\cap (D^{\ast }\times \mathbb{C})=Z(W)\cap (D^{\ast }\times D)$ is an $m$-sheeted unramified covering.

Every zero over $D^{\ast }$ lies in $D$ by step 1.1. Let $(x_n,y_n)$ be any sequence of zeros of the chosen representative with $x_n\to 0$, allowing $x_n=0$; for all sufficiently large $n$, $x_n\in V$, and when $x_n=0$ one has $y_n=0$ by [F1]. Thus the tail of $(y_n)$ lies in the closed disc $\bar{D}$. For any convergent subsequence $y_{n_k}\to y_{\mathrm{\infty }}$, continuity of the polynomial $W$ on a neighbourhood of $\{0\}\times \bar{D}$ gives $W(0,y_{\mathrm{\infty }})={\lim }_{k}W(x_{n_k},y_{n_k})=0$, so [F1] gives $y_{\mathrm{\infty }}^m=0$ and $y_{\mathrm{\infty }}=0$, even if the limit was initially allowed to lie on $\mathrm{\partial }D$. If $y_n$ did not tend to $0$, a subsequence bounded away from $0$ would have a convergent subsequence in $\bar{D}$ with nonzero limit, a contradiction. Hence $y_n\to 0$.

Suppose for contradiction that the total space $F:=Z(W)\cap (D^{\ast }\times D)$ is disconnected, so that $F=F_1\bigsqcup F_2$ with $F_1,F_2$ nonempty, open and closed in $F$.

The function $k(x):=\mathrm{\mid }{F}_1\cap {\pi }^{-1}(x)\mathrm{\mid }$ is locally constant on $D^{\ast }$: if $U\subseteq D^{\ast }$ is a disc over which the covering trivialises with sheets $V_1,\dots ,V_m$, then each $V_i$ is connected by [F10], and $F_1\cap V_i$ is open and closed in $V_i$ because $F_1$ is open and closed in $F$; hence $V_i\subseteq F_1$ or $V_i\cap F_1=\mathrm{\varnothing }$ for each $i$, so $k$ is constant on $U$. As $D^{\ast }$ is connected, $k$ is constant, say $k(x)=k$ for all $x\in D^{\ast }$, and $1\le k\le m-1$ because $F_1$ and $F_2$ are nonempty.

On such a disc $U$ the sheets of $F_1$ are graphs of holomorphic functions ${\phi }_i:U\to D$ by [F6]; define $W_1(x,y):={\prod }_{i\in I}(y-{\phi }_i(x))$ on $U\times D$, where $I$ is the set of sheets contained in $F_1$, so $W_1$ is monic of degree $k$ in $y$ with holomorphic coefficients on $U$. For two discs $U,U^{\mathrm{\prime }}$ the definitions agree on $U\cap U^{\mathrm{\prime }}$, because at each $x\in U\cap U^{\mathrm{\prime }}$ both are the monic degree-$k$ polynomial in $y$ whose $k$ roots are the distinct points of $F_1$ over $x$ by [F3] and [F4]; hence $W_1$ is a well-defined holomorphic function on $D^{\ast }\times D$, monic of degree $k$ in $y$. Defining $W_2$ in the same way from $F_2$, we get a monic holomorphic function of degree $m-k$ on $D^{\ast }\times D$ with $k+(m-k)=m$.

For every $x\in D^{\ast }$, the two monic polynomials $W_1(x,\cdot )W_2(x,\cdot )$ and $W(x,\cdot )$ in $y$ have the same $m$ distinct roots, hence are equal; therefore $W=W_1{W}_2$ on $D^{\ast }\times D$.

The coefficients of $W_1$ and $W_2$ are, up to sign, the elementary symmetric functions of the corresponding root values; by step 1.1 all roots lie in the bounded disc $D$, so every coefficient is a bounded holomorphic function on the punctured disc $D^{\ast }$, and [F7] extends each coefficient holomorphically across the puncture.

The extended coefficients of $W_1$ and $W_2$ vanish at $x=0$: the roots of $W_1$ and of $W_2$ all tend to $0$ as $x\to 0$ by step 2.1, and the elementary symmetric functions are continuous in the roots, so each coefficient has limit $0$. Hence the extensions are Weierstrass polynomials of degrees $k\ge 1$ and $m-k\ge 1$.

The functions $W,W_1,W_2$ are holomorphic on the polydisc $V\times D$ and satisfy $W=W_1{W}_2$ on the nonempty open subset $D^{\ast }\times D$, so by [F8] the identity holds on $V\times D$; hence $W=W_1{W}_2$ in ${\mathcal{O}}_{{\mathbb{C}}^2,0}$ with $W_1,W_2$ Weierstrass polynomials of positive degree.

Step 8.1 gives a factorization of $W$ into positive-degree Weierstrass polynomials. The implication recorded in [F9] makes this a nontrivial germ factorization, contradicting the assumed irreducibility of $W$; therefore $F$ is connected.

By step 1.1 the local covering from step 9.1 is the full zero set $Z(W)\cap (D^{\ast }\times \mathbb{C})$; it is connected and $m$-sheeted, unramified by step 1.1, and every sequence of its zeros whose base coordinates tend to $0$ has fibre coordinates tending to $0$ by step 2.1.