Analytic-boundary exhaustion of a plane domain

Fix $\mathrm{\Omega }$ and the finite set $A$. If $\mathrm{\Omega }=\mathbb{C}$, put $R_0:=1+\sup \{\mathrm{\mid }a\mathrm{\mid }:a\in A\}$ (and $R_0:=1$ when $A=\mathrm{\varnothing }$) and $D_j:=D(0,R_0+j)$ for $j\ge 1$: then $A\subseteq D_1$, the discs increase with $j$, each boundary $\{\mathrm{\mid }z\mathrm{\mid }=r\}$ is the zero set of the real-analytic polynomial $F(z)=\mathrm{\mid }z{\mathrm{\mid }}^2-r^2$ whose gradient $2z$ does not vanish there, so after relabelling the axes [F11] exhibits that zero set near each of its points as the graph of a real-analytic function $g$ with the disc $D(0,r)=\{F<0\}$ equal to one of the two local sides; hence each $\mathrm{\partial }{D}_j$ is real-analytic regular in the one-sided sense of the Statement, and every compact $K\subseteq \mathbb{C}$ is bounded, hence lies in $D_j$ for all large $j$. So assume from now on that $\mathrm{\Omega }\mathrm{\ne }\mathbb{C}$, in which case $\mathbb{C}\setminus \mathrm{\Omega }$ is a nonempty closed set.

List as $Q_1,Q_2,\dots$ the closed discs $\overline{D(c,r)}$ with $c\in \mathbb{Q}[i]$, $r$ a positive rational and $\overline{D(c,r)}\subset \mathrm{\Omega }$, in increasing order of the index of the datum $(c,r)$ in the enumeration of [F1]. Every compact $K\subseteq \mathrm{\Omega }$ is covered by finitely many of the $Q_i$: for $z\in K$ openness gives $\delta (z):=d(z,\mathbb{C}\setminus \mathrm{\Omega })>0$ by [F6], and [F1] together with [F2] supplies $c\in \mathbb{Q}[i]$ and a positive rational $r$ with $\mathrm{\mid }c-z\mathrm{\mid }<\delta (z)\mathrm{/}4$ and $\delta (z)\mathrm{/}4<r<\delta (z)\mathrm{/}2$; every $w$ with $\mathrm{\mid }w-c\mathrm{\mid }\le r$ then has $\mathrm{\mid }w-z\mathrm{\mid }<3\delta (z)\mathrm{/}4$ and therefore $d(w,\mathbb{C}\setminus \mathrm{\Omega })>\delta (z)\mathrm{/}4>0$ by [F6], so $\overline{D(c,r)}\subset \mathrm{\Omega }$ while $z\in D(c,r)$. The interiors of the listed discs thus cover the compact set $K$, and compactness extracts a finite subcover, whose largest index we call $N$.

For every nonempty compact $K\subseteq \mathrm{\Omega }$ the number $\delta (K):=d(K,\mathbb{C}\setminus \mathrm{\Omega })$ is positive: by [F6] the continuous function $z\mapsto d(z,\mathbb{C}\setminus \mathrm{\Omega })$ attains over $K$ its minimum, which is $\delta (K)$, at some $z_{\ast }\in K$, and $\delta (K)=0$ would put $z_{\ast }$ in $\mathrm{\Omega }\cap \overline{\mathbb{C}\setminus \mathrm{\Omega }}=\mathrm{\varnothing }$. Moreover $d(z,\mathbb{C}\setminus \mathrm{\Omega })\ge \delta (K)-d(z,K)$ for every $z\in \mathbb{C}$, by [F6] and the defining infimum of $d(z,K)$.

Let $x_0$ be the least-indexed point of $\mathbb{Q}[i]$ lying in $\mathrm{\Omega }$, which exists by [F1] and [F2] because $\mathrm{\Omega }$ is nonempty and open, and let $c_1$ be the centre of $Q_1$. For each fixed endpoint $p\in A\cup \{c_1\}$, [F3] supplies a polygonal path in $\mathrm{\Omega }$ from $x_0$ to $p$. Its compact image has positive distance from $\mathbb{C}\setminus \mathrm{\Omega }$: the distance function is continuous and positive on that compact subset of the open set $\mathrm{\Omega }$, so it attains a positive minimum by [F5] and [F6]. Perturbing each non-endpoint vertex by less than half that distance keeps every segment in $\mathrm{\Omega }$, because each corresponding point on a perturbed segment moves by at most the maximum endpoint perturbation. Density of $\mathbb{Q}[i]$ therefore gives a path with the same endpoints and all intermediate vertices in $\mathbb{Q}[i]$. For each of these finitely many fixed endpoints, [F1] selects the least-indexed such path from the countable family of finite tuples of rational intermediate vertices; the endpoints, including arbitrary points of $A$, remain fixed. Let $K_1$ be the union of these paths with $Q_1$. Then $K_1$ is a nonempty compact connected subset of $\mathrm{\Omega }$ with $A\subseteq K_1$: each path is a continuous image of a compact interval, hence compact and connected by [F4] and [F5], the disc $Q_1$ is compact and connected by [F4] and [F5], and every member of the union meets the fixed connected member that is the path from $x_0$ to $c_1$, so [F7] applies.

Fix an integer $j\ge 1$ and a nonempty compact connected set $K_j\subseteq \mathrm{\Omega }$ with $A\subseteq K_j$; the base case $K_1$ is supplied by step 1.4. Put ${\delta }_j:=\delta (K_j)>0$ by step 1.3 and let ${\chi }_j:[0,\mathrm{\infty })\to [-1,1]$ be the continuous function ${\chi }_j(t):=\max \{-1,\min \{1,1-\frac{8}{{\delta }_j}(t-\frac{{\delta }_j}{4})\left\}\right\},f_j(z):={\chi }_j(d(z,K_j))\mathrm{.}$ Then ${\chi }_j=1$ on $[0,{\delta }_j\mathrm{/}4]$ and ${\chi }_j=-1$ on $[{\delta }_j\mathrm{/}2,\mathrm{\infty })$, so $f_j$ is continuous with $f_j=1$ on $\{z:d(z,K_j)<{\delta }_j\mathrm{/}4\}$ and $f_j=-1$ on $\{z:d(z,K_j)\ge {\delta }_j\mathrm{/}2\}$.

Let $M_j\ge 1$ be the least integer with $\{z:d(z,K_j)\le {\delta }_j\mathrm{/}2\}\subseteq D(0,M_j\mathrm{/}2)$; such an integer exists because that set is bounded, since compactness gives $B_j^{\ast }:={\max }_{w\in K_j}\mathrm{\mid }w\mathrm{\mid }<\mathrm{\infty }$, and a nearest point $w\in K_j$ gives $\mathrm{\mid }z\mathrm{\mid }\le \mathrm{\mid }w\mathrm{\mid }+\mathrm{\mid }z-w\mathrm{\mid }\le B_j^{\ast }+{\delta }_j\mathrm{/}2$ by [F5] and [F6]. The real-coefficient polynomials in the two coordinates form a unital real subalgebra of $C(\overline{D(0,M_j)},\mathbb{R})$ containing both coordinate functions, hence separating points, so by [F8] some real-coefficient polynomial $P$ satisfies ${\sup }_{\overline{D(0,M_j)}}\mathrm{\mid }P-f_j\mathrm{\mid }<1\mathrm{/}8$. Since $M_j\ge 1$, [F8] lets us approximate the finitely many coefficients of $P$ by rationals so that the resulting rational-coefficient polynomial differs from $P$ by less than $1\mathrm{/}8$ uniformly on this disc. Thus some rational-coefficient polynomial $P_j$ satisfies ${\sup }_{\overline{D(0,M_j)}}\mathrm{\mid }{P}_j-f_j\mathrm{\mid }<1\mathrm{/}4$; take the least-indexed one in the enumeration of [F1]. By [F10] the polynomial $P_j$ is real analytic and $C^{\mathrm{\infty }}$ on the whole plane.

Let $Z_j:=\{z:\mathrm{\mid }z\mathrm{\mid }\le M_j,\mathrm{\nabla }{P}_j(z)=0\}$, compact by [F5] and continuity of $\mathrm{\nabla }{P}_j$, and let $B_j:=P_j(Z_j)$, which is compact by [F5] and null by [F9] because it is a set of critical values of the $C^{\mathrm{\infty }}$ function $P_j$. The interval $(-\frac{1}{2},\frac{1}{2})$ is nondegenerate, so it is not contained in $B_j$ by [F9]; being the complement of the closed set $B_j$ inside an interval, $(-\frac{1}{2},\frac{1}{2})\setminus B_j$ is open and nonempty, so by [F1] and [F2] it contains rationals, and we let $t_j$ be its least-indexed rational point. Consequently $\mathrm{\nabla }{P}_j(z)\mathrm{\ne }0$ whenever $\mathrm{\mid }z\mathrm{\mid }\le M_j$ and $P_j(z)=t_j$, since otherwise $t_j\in B_j$.

Let $D_j$ be the connected component of the open set $\{z:P_j(z)>t_j\}$ containing $K_j$, which exists because $K_j$ is connected and $K_j\subseteq \{P_j>t_j\}$: on $K_j$ we have $d(\cdot ,K_j)=0<{\delta }_j\mathrm{/}4$, so $f_j=1$ there by step 2.1, hence $P_j>3\mathrm{/}4>t_j$ by steps 3.1 and 4.1. Thus $D_j$ is a nonempty open connected set with $K_j\subseteq D_j$, the inclusion by maximality of components.

$D_j\subseteq D(0,M_j)$. Every point $w$ with $\mathrm{\mid }w\mathrm{\mid }=M_j$ has $d(w,K_j)>{\delta }_j\mathrm{/}2$, because $\{z:d(z,K_j)\le {\delta }_j\mathrm{/}2\}\subseteq D(0,M_j\mathrm{/}2)$ by step 3.1; hence $f_j(w)=-1$ by step 2.1 and $P_j(w)<-3\mathrm{/}4<t_j$ by steps 3.1 and 4.1. So the circle $\{\mathrm{\mid }z\mathrm{\mid }=M_j\}$ is disjoint from $\{P_j>t_j\}$, and the connected set $D_j$, which contains $K_j\subseteq D(0,M_j\mathrm{/}2)$, lies in the component $D(0,M_j)$ of the complement of that circle.

$D_j\subseteq \mathrm{\Omega }$, and $\overline{D_j}$ is a compact subset of $\mathrm{\Omega }$. Let $z\in D_j$. If $d(z,K_j)\ge {\delta }_j\mathrm{/}2$ then $f_j(z)=-1$ by step 2.1 and $P_j(z)<-3\mathrm{/}4<t_j$ by steps 3.1 and 4.1, contradicting $z\in D_j$; hence $d(z,K_j)<{\delta }_j\mathrm{/}2$, and by step 1.3 and [F6] $d(z,\mathbb{C}\setminus \mathrm{\Omega })\ge {\delta }_j-d(z,K_j)>{\delta }_j\mathrm{/}2>0$, so $z\in \mathrm{\Omega }$. Passing to closures, [F6] gives $\overline{D_j}\subseteq \{z:d(z,K_j)\le {\delta }_j\mathrm{/}2\}$, and that set is closed, bounded by step 3.1 and contained in $\mathrm{\Omega }$ by the same distance inequality, hence compact by [F5].

$\mathrm{\partial }{D}_j\subseteq \{z:P_j(z)=t_j\}$, and $\mathrm{\nabla }{P}_j\mathrm{\ne }0$ at every point of $\mathrm{\partial }{D}_j$. Let $z_0\in \mathrm{\partial }{D}_j$. Since $D_j\subseteq \{P_j>t_j\}$ and $P_j$ is continuous, while $z_0$ lies in the closure of $D_j$, we get $P_j(z_0)\ge t_j$; and $z_0\in \overline{D_j}\subseteq D(0,M_j)$ by steps 6.1 and 6.2. If $P_j(z_0)>t_j$, then $\{P_j>t_j\}$ contains a disc $B$ around $z_0$; $B$ is connected by [F4] and meets $D_j$ because $z_0$ is a boundary point of $D_j$, so $B\subseteq D_j$ by maximality of components, making $z_0$ an interior point of $D_j$ and contradicting $z_0\in \mathrm{\partial }{D}_j$. Hence $P_j(z_0)=t_j$, and $\mathrm{\nabla }{P}_j(z_0)\mathrm{\ne }0$ by step 4.1.

Construction of the next compact set. Let $y_j$ be the least-indexed point of $\mathbb{Q}[i]$ lying in $D_j$, which exists by [F1] and [F2] because $D_j$ is nonempty and open, let $c_{j+1}$ be the centre of $Q_{j+1}$, and let $P_{j+1}$ be the union of the two least-indexed polygonal paths whose intermediate vertices lie in $\mathbb{Q}[i]$, from $x_0$ to $y_j$ and from $x_0$ to $c_{j+1}$. These paths exist by the argument of step 1.4; their endpoints are rational as well. Then $K_{j+1}:=\overline{D_j}\cup Q_{j+1}\cup P_{j+1}$ is a nonempty compact connected subset of $\mathrm{\Omega }$ with $A\subseteq K_{j+1}$, so that the construction of step 2.1 can be applied to it: compactness follows from [F5] and step 6.2, and connectedness from [F7], because $\overline{D_j}$ meets $P_{j+1}$ at $y_j$, while $Q_{j+1}$ meets $P_{j+1}$ at $c_{j+1}$, and each of the three members is connected by [F3], [F4] and step 6.2. Moreover $A\subseteq K_j\subseteq D_j\subseteq K_{j+1}$.

The boundary $\mathrm{\partial }{D}_j$ is real-analytic regular. Fix $z_0\in \mathrm{\partial }{D}_j$. By step 7.1, after relabelling axes, $D_y{P}_j(z_0)\mathrm{\ne }0$. Apply the inverse assertion of [F11] to $H(x,y)=(x,P_j(x,y)-t_j)$, whose Jacobian determinant at $z_0$ is $D_y{P}_j(z_0)$. Restrict its analytic inverse $K$ to a rectangle $V=I\times (-ϵ,ϵ)$ about $H(z_0)$ and put $U=K(V)$. The first coordinate identity forces $K(x,s)=(x,k(x,s))$. Thus the zero set in $U$ is the graph $y=g(x):=k(x,0)$, and the positive and negative sides are respectively $K(I\times (0,ϵ))$ and $K(I\times (-ϵ,0))$. Both are connected, being continuous images of convex rectangles; they are the two components of the complement of the graph in $U$. The positive side meets $D_j$ since $z_0\in \mathrm{\partial }{D}_j$, so maximality of the component $D_j$ puts that whole side in $D_j$. Conversely $D_j\cap U$ lies in that side by its definition. Every graph point is approached by points of the positive side and belongs to neither open side, hence $\mathrm{\partial }{D}_j\cap U$ is exactly the graph. This proves the required one-sided regularity without assuming the sign of $D_y{P}_j$. The same local argument with the negative side applies to the discs in step 1.1.

Applying the construction of steps 2.1 through 5.1 to the admissible set $K_{j+1}$ of step 7.2 produces the connected component $D_{j+1}$ of $\{P_{j+1}>t_{j+1}\}$ containing $K_{j+1}$, so $K_{j+1}\subseteq D_{j+1}$. Since $D_j\subseteq \overline{D_j}\subseteq K_{j+1}$ by step 7.2 and $D_{j+1}$ is a component, maximality of components yields $D_j\subseteq D_{j+1}$.

Every compact $K\subseteq \mathrm{\Omega }$ lies in $D_m$ for all sufficiently large $m$. By step 1.2 there is $N$ with $K\subseteq Q_1\cup \cdots \cup Q_N$. For each $m\ge 1$ the set $Q_m$ satisfies $Q_m\subseteq K_m$ by steps 1.4 and 7.2, and $K_m\subseteq D_m$ by step 5.1 applied to $K_m$, while $D_m\subseteq D_{m^{\mathrm{\prime }}}$ for $m^{\mathrm{\prime }}\ge m$ by step 8.2; hence $K\subseteq D_N\subseteq D_{m^{\mathrm{\prime }}}$ for every $m^{\mathrm{\prime }}\ge N$.

The sequence $D_1\subseteq D_2\subseteq \cdots$ obtained by applying steps 2.1 through 7.2 inductively, starting from $K_1$ of step 1.4, consists of nonempty relatively compact connected open subsets of $\mathrm{\Omega }$ with real-analytic regular boundary by steps 6.2 and 8.1, contains $A$ in $D_1$ because $A\subseteq K_1\subseteq D_1$ by steps 1.4 and 5.1, and exhausts $\mathrm{\Omega }$ in the required sense by step 9.1. Every selection made above is either a finite selection or a least-index selection in one of the fixed at most countable families of [F1], or the least-indexed rational point of a nonempty open set, whose existence is [F2]; no choice principle was used.