---
abstract: |
    Let $\varphi: D\rightarrow \Omega$ be a homeomorphism from a circle domain $D$ onto a domain $\Omega\subset\hat{{\mathbb{C}}}$. We obtain necessary and sufficient conditions (1) for $\varphi$ to have a continuous extension to the closure $\overline{D}$ and (2) for such an extension to be injective. Further assume that $\varphi$ is conformal and that $\partial\Omega$ has at most countably many non-degenerate components $\{P_n\}$ whose diameters have a finite sum $\displaystyle\sum_n{\rm diam}(P_n)<\infty$. When the point components of $\partial D$ or those of $\partial \Omega$ form a set of $\sigma$-finite linear measure, we can show that $\varphi$ continuously extends to $\overline{D}$ if and only if all the components of $\partial\Omega$ are locally connected. This generalizes Carathéodory’s Continuity Theorem, that concerns the case when $D$ is the open unit disk $\left\{z\in\hat{{\mathbb{C}}}: |z|<1\right\}$, and allows us to derive a new generalization of the Osgood-Taylor-Caratheodry Theorem.

    0.5cm

    **Keywords.** *Carathéodory’s Continuity Theorem, Peano Compactum, generalized Jordan domain*

    **MSC 2010: Primary 30A72, 30D40, Secondary 54C20, 54F25.**
author:
- Jun Luo
- 'Xiao-Ting Yao'
title: 'To Generalize Carathéodory’s Continuity Theorem[^1]'
---

Introduction and What We Study
==============================

There are two questions that are of particular interest from a topological viewpoint. In the first, we want to decide whether two spaces $X$ and $Y$ are topologically equivalent or homeomorphic, in the sense that there is a homeomorphism $h_1:X\rightarrow Y$. In the second, the spaces $X$ and $Y$ are respectively embedded in two larger spaces, say $\hat{X}$ and $\hat{Y}$, and we wonder whether a continuous map $h_2:X\rightarrow Y$ allows a continuous extension $\hat{h}_2: \hat{X}\rightarrow\hat{Y}$. Our study concerns a special case of the second question, when $X$ is a circle domain and $h_2$ a conformal homeomorphism sending $X$ onto a domain $Y\subset\hat{{\mathbb{C}}}$. In such a case $X$ and $Y$ are said to be [**conformally equivalent**]{}.

Our major aim in this paper is to generalize Carathëodory’s Continuity Theorem [@Caratheodory13-a]. See also [@Arsove68-a Theorem 3] or [@Pom92 p.18].

A conformal homeomorphism $\varphi:{\mathbb{D}}\rightarrow \Omega\subset\hat{{\mathbb{C}}}$ of the unit disk ${\mathbb{D}}=\{z: |z|<1\}$ has a continuous extension $\overline{\varphi}: \overline{{\mathbb{D}}}\rightarrow\overline{\Omega}$ if and only if the boundary $\partial\Omega$ is a Peano continuum, [*i.e.*]{} a continuous image of the interval $[0,1]$.

If $\Omega$ in the above theorem is a [**Jordan domain**]{}, so that its boundary is a [**Jordan curve**]{}, the extension $\overline{\varphi}: \overline{{\mathbb{D}}}\rightarrow\overline{\Omega}$ is actually injective. This has been obtained earlier by Osgood and Taylor [@Osgood-Taylor1913 Corollary 1] and independently by Carathéodory [@Caratheodory13-b]. It will be referred to as the Osgood-Taylor-Carathéodory Theorem. See for instance [@Arsove68-a Theorem 4]. Here we also call it shortly the OTC Theorem.

A conformal homeomorphism $\varphi:{\mathbb{D}}\rightarrow \Omega\subset\hat{{\mathbb{C}}}$ has a continuous and injective extension to $\overline{{\mathbb{D}}}$ if and only if the boundary $\partial\Omega$ is a simple closed curve.

There are very recent generalizations of the above OTC Theorem. See [@He-Schramm93 Theorem 3.2], [@He-Schramm94 Theorem 2.1], and [@Ntalampekos-Younsi19 Theorem 6.1]. Those generalizations are closely connected with a very famous example of the first question, proposed in 1909 by Koebe [@Koebe09].

Is every domain $\Omega\subset\hat{{\mathbb{C}}}$ conformally equivalent to a circle domain ?

When $\Omega$ is finitely connected, in the sense that its boundary has finitely many components, the above question is resolved by Koebe [@Koebe18]. See the following theorem. The special case when $\Omega$ is simply connected is discussed in the well known Riemann Mapping Theorem.

Each finitely connected domain $\Omega\subset\hat{{\mathbb{C}}}$ is conformally equivalent to a circle domain $D$, unique up to Möbius transformations.

When $\Omega$ is at most countably connected, He and Schramm [@He-Schramm93] obtained the same result.

Each countably connected domain $\Omega\subset\hat{{\mathbb{C}}}$ is conformally equivalent to a circle domain, unique up to Möbius transformations.

This covers some earlier and more resticted results that partially solve [**Koebe’s Question**]{}, when additional conditions on a countably connected domain $\Omega$ are assumed. Among others, one may see [@Strebel51] for such a result. A slightly more general version of the above theorem, on almost circle domains, is given by He and Schramm in [@He-Schramm95a]. Here $\Omega\subset A$ is a relative circle domain in $\Omega$ provided that each component of $A\setminus\Omega$ is either a point or a closed geometric disk. An equivalent statement, pointed out by He and Schramm in [@He-Schramm95a], reads as follows.

Given a countably connected domain $A\subset\hat{{\mathbb{C}}}$, every relative circle domain $\Omega\subset A$ is conformally equivalent to a circle domain $D$, unique up to Möbius transformations.

The uniqueness part of the above extended versions of [**Koebe’s Theorem**]{} comes from the conformal rigidity of specific circle domains. For circle domains that are at most countably connected and even for those that have a boundary with $\sigma$-finite linear measure, the conformal rigidity is known. See [@He-Schramm93 Theorem 3.1] and [@He-Schramm94]. To obtain the conformal rigidity of the underlying circle domains, He and Schramm actually employ some extended version of the OTC Theorem. See [@He-Schramm93 Theorem 3.2] for the case of countably connected domains. See [@He-Schramm95a Lemma 5.3] and [@He-Schramm95a Theorem 6.1] for the case of almost circle domains.

Before addressing on what we study, we recall that in Carathéodory’s Continuity Theorem, the “only if” part follows from very basic observations. On the other hand, the “if” part may be obtained by using the prime ends of $\Omega$, or equivalently, the cluster sets of $\varphi$. See [@Caratheodory13-a] and [@CL66] for the theory of prime ends and for that of cluster sets. Moreover, by the Hahn-Mazurkewicz-Sierpiński Theorem [@Kuratowski68 p,256, $\S50$, II, Theorem 2], a compact connected metric space is a Peano continuum if and only if it is locally connected. Therefore, in Carathéodory’s Continuity Theorem one may replace the property of being a Peano continuum with that of being locally connected. In such a form, the same result still holds, if we change ${\mathbb{D}}$ into a circle domain that is finitely connected, [*i.e.*]{}, having finitely many boundary components.

We will characterize all homeomorphisms $\varphi: D\rightarrow\Omega$ of an arbitrary circle domain $D$ onto a domain $\Omega\subset\hat{{\mathbb{C}}}$ that allow a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ to the closure $\overline{D}$. We also analyse the restriction of $\overline{\varphi}$ to any boundary component of $D$, trying to find conditions for such a restriction to be injective. More importantly, we will find answers to the following.

Under what conditions does $\varphi$ extend continuously to $\overline{D}$, if it is further assumed to be a conformal map ?

What We Obtain and What Are Known
=================================

In the first theorem we find a topological counterpart for Carathéodory’s Continuity Theorem.

\[topological-cct\] Any homeomorphism $\varphi$ of a generalized Jordan domain $D$ onto a domain $\Omega\subset\hat{{\mathbb{C}}}$ has a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ if and only if the conditions below are both satisfied.

-   The boundary $\partial\Omega$ is a Peano compactum.

-   The oscillations of $\varphi$ satisfy $\displaystyle\underline{\lim}_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial D$.

A [**Peano compactum**]{} means a compact metrisable space whose components are each a Peano continuum such that for any $C>0$ at most finitely many of the components are of diameter $>C$. A [**generalized Jordan domain**]{} is defined to be a domain $\Omega\subset\hat{{\mathbb{C}}}$ whose boundary $\partial\Omega$ is a Peano compactum, such that all the components of $\partial\Omega$ are each a point or a Jordan curve. And, for any $r>0$ and any point $z_0\in\partial D$, the oscillation of $\varphi$ at $C_r(z_0)\cap D$ is $\sigma_r(z_0)=\sup\{|\varphi(x)-\varphi(y)|: x,y\in D, |x-z|=|y-z|=r\}$. Here $C_r(z_0)=\{z: |z-z_0|=r\}$.

The same philosophy has been employed by Arsove [@Arsove68-a]. Indeed, the result of Theorem \[topological-cct\] for simply connected $D$ is known [@Arsove68-a Theorem 1]. In the same work, Arsove also gives a topological counterpart for the OTC Theorem [@Arsove68-a Theorem 2]. In the next theorem,, we continue to obtain a topological counterpart for generalized Jordan domains in the second theorem.

\[topological-otc\] Any homeomorphism $\varphi$ of a generalized Jordan domain $D$ onto a domain $\Omega\subset\hat{{\mathbb{C}}}$ has a continuous injective extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ if and only if the conditions below are satisfied:

-   The domain $\Omega$ is a generalized Jordan domain.

-   The oscillations of $\varphi$ satisfy $\displaystyle\underline{\lim}_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial D$,

-   No arc on $\partial D$ of positive length is sent by $\overline{\varphi}$ to a single point of $\partial\Omega$.

In the above theorems the homeomorphism $\varphi$ is not required to be conformal. When this is assumed and $D$ is a circle domain, three special cases are already known in which $\varphi$ extends to be a homeomorphism between $\overline{D}$ and $\overline{\Omega}$. See [@He-Schramm93 Theorem 3.2], [@He-Schramm94 Theorem 2.1], and [@Ntalampekos-Younsi19 Theorem 6.1]. In each of these cases, the circle domain $D$ is required to have a boundary with $\sigma$-finite linear measure or to satisfy a quasi-hyperbolic condition, while $\Omega$ is either a circle domain or a generalized Jordan domain that is [*cofat*]{} in Schramm’s sense, so that all its complementary components are each a single point or closed Jordan domain that is not far from a geometric disk. When both $D$ and $\Omega$ are required to be generalized Jordan domains that are countably connected and cofat, any conformal homeomorphism $\varphi: D\rightarrow\Omega$ extends to be a homeomorphism between $\overline{D}$ and $\overline{\Omega}$ provided that the boundary map $\varphi^B$ gives a bijection between the point components of $\partial D$ and those of $\partial\Omega$. See [@Schramm95 Theorem 6.2].

Removing the requirement of cofatness, we will find new conditions for an arbitrary conformal homeomorphism $\varphi: D\rightarrow\Omega$ to extend continuously to the closure $\overline{D}$. This extends Carathéodory’s Continuity Theorem to infinitely connected circle domains and leads us to a new generalization of the OTC Theorem. Such a generalization has overlaps with but is not covered by any of the known extended versions of the OTC Theorem, that have been obtained in [@He-Schramm93; @He-Schramm94; @Schramm95; @Ntalampekos-Younsi19].

Recall that, by Theorem \[topological-cct\](1), we may confine ourselves to the case that the boundary $\partial\Omega$ is a Peano compactum. Therefore, in the third theorem we characterize all domains $\Omega\subset\hat{{\mathbb{C}}}$ such that the boundary $\partial\Omega$ is a Peano compactum.

\[topology\_metric\] Each of the following is necessary and sufficient for an arbitrary domain $\Omega\subset\hat{{\mathbb{C}}}$ to have its boundary being a Peano compactum:

\(1) $\Omega$ has property S,

\(2) every point of $\partial \Omega$ is locally accessible,

\(3) every point of $\partial \Omega$ is locally sequentially accessible,

\(4) $\Omega$ is finitely connected at the boundary, and

\(5) the completion of $\Omega$ under the diameter distance is compact.

On the one hand, Theorem \[topology\_metric\] demonstrates an interplay between the topology of $\Omega$, that of the boundary $\partial\Omega$, and the completion of the metric space $(\Omega,d)$. Here $d$ denotes the diameter distance, which is also called the Mazurkiewicz distance. See [@Herron12] for a special sub-case of the above Theorem \[topology\_metric\], when $\Omega$ is assumed to be simply connected. On the other, Theorem \[topology\_metric\] is also motivated by and actually provides a generalization for a fundamental characterization of planar domains that have property $S$. See for instance [@Whyburn42 p.112, Theorem (4.2)], which will be cited wholly in this paper and is to appear as Theorem \[whyburn\_112\] (in Section \[proof\_1/2\] of this paper).

Note that the completion of $(\Omega,d)$ is compact if and only if $\Omega$ is finitely connected at the boundary [@BBS16 Theorem 1.1]. The authors of [@BBS16] also obtain the equivalences between (2), (4) and (5) for countably connected domains $\Omega\subset\hat{{\mathbb{C}}}$ [@BBS16 Theorem 1.2] or slightly more general choices of $\Omega$ [@BBS16 Theorem 4.4]. The above Theorem \[topology\_metric\] improves these earlier results, by obtaining all these equivalences for an arbitrary planar domain $\Omega$ and relating them to the property of having a boundary that is a Peano compactum.

Now, we are ready to present on two approaches, that are new, to generalize Carathéodory’s Continuity Theorem. To do that, we further suppose that the domain $\Omega$ has at most countably many non-degenerate boundary components $P_n$ whose diameters satisfy $\sum_n{\rm diam}(P_n)<\infty$. For the sake of convenience, a domain $\Omega$ satisfying the above inequality $\displaystyle\sum_n{\rm diam}(P_n)<\infty$ concerning the diameters of its non-degenerate boundary components will be called a domain with [**diameter control**]{}.

By the first approach, we obtain the following.

\[arsove\] Let $\Omega\subset\hat{{\mathbb{C}}}$ be a domain with countably many non-degenerate boundary components $P_n$ such that the sum of diameters $\sum_n{\rm diam}(P_n)$ is finite. Suppose that the linear measure of $\displaystyle\partial\Omega\setminus\bigcup_nP_n$ is $\sigma$-finite. Then any conformal homeomorphism $\varphi:D\rightarrow \Omega$ from a circle domain $D$ onto $\Omega$ has a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ if and only if $\partial\Omega$ is a Peano compactum.

In the second approach, we require instead that the point components of $\partial D$ form a set of $\sigma$-finite linear measure. This happens if and only if the whole boundary $\partial D$ has a $\sigma$-finite linear measure. In other words, we have the following.

\[arsove\_sigma\] Let $\Omega\subset\hat{{\mathbb{C}}}$ be a domain with diameter control, so that $\partial\Omega$ has at most countably many non-degenerate boundary components $P_n$ satisfying $\sum_n{\rm diam}(P_n)<\infty$. Let $D$ be a circle domain whose boundary has $\sigma$-finite linear measure. Then any conformal homeomorphism $\varphi:D\rightarrow \Omega$ has a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ if and only if $\partial\Omega$ is a Peano compactum.

Note that, in Theorems \[arsove\] and \[arsove\_sigma\], the continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ exists if and only if one of the five conditions given in Theorem \[topology\_metric\] is satisfied.

Among others, Theorem \[arsove\_sigma\] has motivations from a recent work by He and Schramm [@He-Schramm94]. This works centers around the conformal rigidity of circle domains that have a boundary with $\sigma$-finite linear measure. Particularly, in the proof for [@He-Schramm94 Theorem 2.1] we find detailed techniques that are very useful in our study. He and Schramm [@He-Schramm94] consider conformal homeomorphisms between circle domains, while in Theorem \[arsove\_sigma\] we study conformal homeomorphisms from a circle domain $D$ onto a general planar domain $\Omega$. Note that the inequalities obtained in [@He-Schramm94 Lemmas 1.1 and 1.2 and 1.4] are among the crucial elements that constitute the proof for [@He-Schramm94 Theorem 2.1]. In order to obtain these inequalities, one needs to assume at least that the complementary components of $\Omega$ are $L$-nondegenerate for some constant $L>0$. Such domains are also called [**cofat domains**]{} in [@Ntalampekos-Younsi19] and in [@Schramm95].

Instead of assuming the property of being cofat, we focus on domains $\Omega$ with diameter control. This is the major difference between Theorem \[arsove\_sigma\] and the earlier results obtained in [@He-Schramm94; @Ntalampekos-Younsi19; @Schramm95]. For this flexibility, to choose $\Omega$ more freely among a large family of planar domains, we pay a price by assuming in addition the [**diameter control**]{}, so that $\Omega$ has at most countably many components whose diameters have a finite sum $\sum_n{\rm diam}(P_n)<\infty$. Note that in the cofat situation, there is a natural inequality $\sum_n\left({\rm diam}(P_n)\right)^2<\infty$, ensured by the fact that every domain on the sphere has a finite area.

Theorems \[arsove\] and \[arsove\_sigma\] may be slightly improved by replacing $D$ with a generalized circle domain. See Theorems \[arsove-new\] and \[arsove\_sigma-new\]. From this we can infer a new generalization of the OTC Theorem. Such a generalization has overlaps with and is not covered by any of the earlier ones obtained in [@He-Schramm93 Theorem 3.2], [@He-Schramm94 Theorem 2.1], [@Ntalampekos-Younsi19 Theorem 1.6], and [@Schramm95 Theorem 6.2]. The original form of the OTC Theorem is about a conformal homeomorphism between two Jordan domains. In the next theorem, we extend the OTC Theorem to conformal homeomorphisms between two generalized Jordan domains with [**diameter control**]{}.

\[OTC-b\] Given a conformal map $h: D\rightarrow \Omega$ between two generalized Jordan domains, such that both $\partial D$ and $\partial\Omega$ have at most countably many non-degenerate components, say $\{Q_n\}$ and $\{P_n\}$, whose diameters have a finite sum $\sum_n{\rm diam}P_n+\sum_n{\rm diam}(Q_n)<\infty$. Suppose that the point components of $\partial D$ or those of $\partial\Omega$ form a set of $\sigma$-finite linear measure. Then $\varphi$ extends to be a homeomorphism from $\overline{D}$ onto $\overline{\Omega}$.

The other parts of our paper are arranged as follows.

In section \[proof\_1/2\] we prove Theorems \[topological-cct\] and \[topological-otc\]. To do that, we firstly establish in subsection \[s-pc\] a connection between the topology of a planar domain $\Omega$ and that of its boundary $\partial\Omega$, showing that $\Omega$ has property $S$ if and only if $\partial\Omega$ is a Peano compactum. See Theorem \[property\_s\]. Then we discuss in subsection \[cluster\_set\] continuous function of a generalized Jordan domain and show that all the cluster sets of such a function are connected. See Theorem \[connected\_cluster\]. In this subsection, we also provide a non-trivial characterization of generalized Jordan domain. See Theorem \[jordan\]. Then, in subsection \[outline\] and in subsection \[outline-otc\], we respectively prove Theorems \[topological-cct\] and \[topological-otc\]. In section \[topology\], we prove Theorems \[topology\_metric\].

In section \[outline-1\] we firstly discuss a special case of Theorem \[arsove\], when the point components of $\partial\Omega$ form a set of zero linear measure. See Theorem \[sufficient\]. Then we use very similar arguments, with necessary adjustments and more complicated details, to construct a proof for Theorem \[arsove\].

In section \[outline-2\] we will prove Theorem \[arsove\_sigma\], when the point components of $\partial D$ form a set of zero linear measure. The proofs for this theorem and Theorem \[arsove\] are both based on an estimate of the oscillations for some conformal homeomorphism $\varphi: D\rightarrow\Omega$ of a circle domain $D$, so that Theorem \[topological-cct\] may be applied. Note that the results for Theorems \[arsove\] and \[arsove\_sigma\] still hold, even if the circle domain $D$ is replaced by a generalized Jordan domain. See Theorems \[arsove-new\] and \[arsove\_sigma-new\].

Finally, in section \[final\] we will prove Theorem \[OTC-b\]. Here we also recall earlier results that provide generalized versions of the classical OTC Theorem. See Theorems \[OTC-countable\] to \[OTC-3\]. These results arise very recent studies that provide the latest partial solutions to Koebe’s Question. They are comparable with Theorem \[OTC-b\], especially Theorem \[OTC-3\].

To Extend Homeomorphisms on a Circle Domain {#proof_1/2}
===========================================

The target of this section is to prove Theorems \[topological-cct\] and \[topological-otc\].

To do that, we need a result that connects the topology of a planar domain $\Omega\subset\hat{{\mathbb{C}}}$ to that of its boundary, stating that $\Omega$ has property $S$ if and only if $\partial\Omega$ is a Peano compactum. We also need to analyze the cluster sets of a homeomorphism $h$, possibly not conformal, that sends a generalized Jordan domain $D$ onto a planar domain $\Omega$. Then we will be ready to construct the proofs for Theorems \[topological-cct\] and \[topological-otc\].

All these materials are presented separately in the following four subsections.

Property $S$ and the property of being a Peano Compactum {#s-pc}
--------------------------------------------------------

The property $S$ for planar domains and the property of being a Peano compactum, for compact planar sets, are closely connected. Such a connection is motivated by and provides a partial generalization for [@Whyburn42 p.112, Theorem (4.2)], which reads as follows.

\[whyburn\_112\] If $\Omega\subset{\mathbb{C}}$ is a region whose boundary is a continuum the following are equivalent:

-   that $\Omega$ have Property $S$,

-   that every point of $\partial\Omega$ be regularly accessible from $\Omega$,

-   that every point of $\partial\Omega$ be accessible from all sides from $\Omega$,

-   that $\partial\Omega$ be locally connected, or equivalently, a Peano continuum.

Here a region is a synonym of a domain and a metric space $X$ is said to have Property $S$ provided that for each $\epsilon>0$ the set $X$ is the union of finitely many connected sets of diameter less than $\epsilon$ [@Whyburn42 p.20]. Also, note that a point $p\in\partial\Omega$ is said to be [*regularly accessible from $\Omega$*]{} provided that for any $\epsilon>0$ there is a number $\delta>0$ such that for any $x\in\Omega$ with $|x-p|<\delta$ one can find a simple arc $\overline{xp}\subset \Omega\cup\{p\}$ that joins $x$ to $p$ and has a diameter $<\epsilon$ [@Whyburn42 p.111]. Note that a point $x\in\partial\Omega$ regularly accessible is also said to be [*locally accessible*]{} [@Arsove67].

The above theorem provides another motivation for Theorem \[property\_s\] that is of its own interest. We find a partial generalization for it, keeping items (ii) and (iii) untouched for the moment.

\[property\_s\] A domain $\Omega\subset\hat{{\mathbb{C}}}$ has Property $S$ if and only if $\partial\Omega$ is a Peano compactum.

When proving Theorem \[property\_s\] we will use two notions introduced in [@LLY-2019], the [**Schönflies condition**]{} and the [**Schönflies relation**]{} for planar compacta.

\[Schonflies\_condition\] A compactum $K\subset{\mathbb{C}}$ satisfies the Schönflies condition provided that for the strip $W=W(L_1,L_2)$ bounded by two arbitrary parallel lines $L_1$ and $L_2$, the [**difference**]{} $\overline{W}\setminus K$ has at most finitely many components intersecting $L_1$ and $L_2$ at the same time.

\[Schonflies\_relation\] Given a compact set $K\subset{\mathbb{C}}$. The Schönflies relation on $K$, denoted as $R_K$, is a reflexive relation such that two points $x_1\ne x_2\in K$ are related under $R_K$ if and only if there are two disjoint simple closed curves $J_i\ni x_i$ such that $\overline{U}\cap K$ has infinitely many components intersecting $J_1, J_2$ both. Here $U$ is the component of  $\hat{{\mathbb{C}}}\setminus(J_1\cup J_2)$ with $\partial U=J_1 \cup J_2$.

By [@LLY-2019 Theorem 3], a compact $K\subset{\mathbb{C}}$ is a Peano compactum if and only if it satisfies the Schönflies condition. On the other hand, by [@LLY-2019 Theorem 7], a compact $K\subset{\mathbb{C}}$ is a Peano compactum if and only if $R_K$ is [**trivial**]{}, so that $(x,y)\in R_K$ indicates $x=y$. These results have motivations from recently developed topological models that are very helpful in the study of polynomial Julia sets. See for instance [@BCO11; @BCO13; @Curry10; @Kiwi04]. It is noteworthy that these models also date back to the 1980’s, when Thurston and Douady and their colleagues started applying [**Carathéodory’s Continuity Theorem**]{} to the study of polynomial Julia sets, which are assumed to be connected and locally connected. See for instance [@Douady93] and [@Thurston09].

We start from a proof by contradiction for the “only if” part.

Suppose on the contrary that $\Omega$ has Property $S$ but $\partial\Omega$ is not a Peano compactum. There would exist two parallel lines $L_1,L_2$ such that for the unbounded strip $W=W(L_1,L_2)$ lying between $L_1$ and $L_2$, the [**difference**]{} $\overline{W}\setminus \partial\Omega$ has infinitely many components intersecting both $L_1$ and $L_2$. Denote those components as $W_1,W_2,\ldots$. Since every $W_i$ is arcwise connected, we may choose simple open arcs $\alpha_i\subset W_i$ joining a point $a_n$ on $\overline{W_i}\cap L_1$ to a point $b_n$ on $\overline{W_i}\cap L_2$. Renaming the arcs $\alpha_n$ if necessary, we may assume that for any $n>1$, the two arcs $\alpha_{n-1}$ and $\alpha_{n+1}$ lie in different components of $W\setminus\alpha_n$. Thus the arcs $\alpha_n$ may be arranged inside $W$ linearly from left to right. See the following figure for a simplified depiction of this arrangement.

-0.25cm

iin [1,...,50]{} [ (3+3\*i/50,5) – (2+3\*i/50,3) – (3.5+3\*i/50,1.5) – (3+3\*i/50,0); (11+3\*i/50,5) – (10+3\*i/50,3) – (11.5+3\*i/50,1.5) – (11+3\*i/50,0); ]{}

(-2,0)–(25,0); (-2,5)–(25,5); (3,5) – (2,3) – (3.5,1.5) – (3,0); (6,5) – (5,3) – (6.5,1.5) – (6,0); (11,5) – (10,3) – (11.5,1.5) – (11,0); (14,5) – (13,3) – (14.5,1.5) – (14,0);

at (25.75,5) [$L_1$]{}; at (25.75,0) [$L_2$]{}; at (1.4,2.9) [$\alpha_1$]{}; at (3.75,2.9) [$D_1$]{}; at (11.75,2.9) [$D_n$]{}; at (6.0,2.9) [$\alpha_2$]{}; at (9.4,2.9) [$\alpha_n$]{}; at (14.5,2.9) [$\alpha_{n+1}$]{}; at (11,5.5) [$a_n$]{}; at (11,-0.5) [$b_n$]{}; at (9.0,1.4) [$\cdots\cdots$]{}; at (16.5,1.4) [$\cdots\cdots$]{};

(11,0) circle (0.1); (11,5) circle (0.1); (14,0) circle (0.1); (14,5) circle (0.1); (6,0) circle (0.1); (6,5) circle (0.1); (3,0) circle (0.1); (3,5) circle (0.1);

-0.75cm

-0.5cm

Let $D_n (n\ge1)$ be the unique bounded component of ${\mathbb{C}}\setminus(L_1\cup L_2\cup \alpha_n\cup\alpha_{n+1})$. Then each $D_n$ is a Jordan domain; moreover, the closed disk $\overline{D_n}$ contains a continuum $M_n\subset\partial\Omega$ that separates $\alpha_n$ from $\alpha_{n+1}$ in $\overline{D_n}$. Such a continuum $M_n$ must intersect both $L_1$ and $L_2$. Therefore, we can choose $x_n\in M_{2n-1}$ for all $n\ge1$ with $${\rm dist}(x_n,L_1)={\rm dist}(x_n,L_2):=\min\left\{\left|x_n-z\right|:\ z\in L_2\right\}.$$ Let $\epsilon>0$ be a number smaller than $\frac14\text{\rm dist}(L_1,L_2)$. Since $x_n\in M_{2n-1}\subset\partial\Omega$ we may find a point $y_n\in \Omega\cap D_{2n-1}$ such that $|x_n-y_n|<\epsilon$. Clearly, for any $m,n\ge1$ the two points $y_n, y_{n+m}\in \Omega$ are separated in $\overline{W}$ by $M_{2n}$. In other words, we have obtained an infinite set $\{y_n\}$ of points in $\Omega$, no two of which may be contained in a single connected subset of $\Omega$ that are of diameter less than $\epsilon$. This leads to a contradiction to the assumption that $\Omega$ has Property $S$.

Then we continue to prove the “if” part. Again we will construct a proof by contradiction.

Suppose on the contrary that $\partial\Omega$ is a Peano compactum but $\Omega$ does not have Property $S$. Then we could find a number $\epsilon>0$ and an infinite set $\{x_i\}$ of points $\Omega$ no two of which lie together in a single connected subset of $\Omega$ having diameter less than $3\epsilon$. By compactness of $\overline{\Omega}$, we may assume that $\lim\limits_{i\rightarrow\infty} x_i=x$. The way we choose the points $x_i$ then implies that $x\in\partial\Omega$. In the following, let $D_r(z)=\{w\in{\mathbb{C}}:\ |z-w|<r\}$ for $r>0$.

Given a number $r\in(0,\epsilon)$, there exists an integer $i_0\ge1$ such that $x_i\in D_r(x)$ for all $i\ge i_0$. Fix a point $x_0\in \Omega$ with $|x-x_0|>\epsilon$ and choose arcs $\alpha_i\subset \Omega$ starting from $x_0$ and ending at $x_i$. Now for any $i\ge i_0$ let $a_i\in\alpha_i$ be the last point at which $\alpha_i$ leaves $\partial D_\epsilon(x)$; let $b_i\in\alpha_i$ be the first point after $a_i$ at which $\alpha_i$ encounters $\partial D_r(x)$. Let $\beta_i$ be the sub-arc of $\alpha_i$ between $a_i$ and $b_i$. Let $\gamma_i$ be the sub-arc of $\alpha_i$ between $b_i$ and $x_i$.

(0,0) circle (5); (0,0) circle (2.5); at (6.25,0) [$\partial D_\epsilon(x)$]{}; at (0.75,-3.00) [$\partial D_r(x)$]{}; at (0.75,0) [$x$]{}; (0,0)circle (0.2); at (-6.75,0) [$x_0$]{}; (-6,0)circle (0.2);

at (0.8,-1.2) [$x_i$]{}; (0,-1.2)circle (0.2);

(-1.5,-2.0)circle (0.2); (-4,-3.0)circle (0.2); at (-4.5,-3.5) [$a_i$]{}; at (-1.35,-1.1) [$b_i$]{}; (-4,-3) – (-2.75,-3) – (-1.5,-2); at (-2.25,-3.5) [$\beta_i$]{};

-0.75cm

Since no two of the points $\{x_i\}$ are contained by a single connected subset of $\Omega$ that is of diameter less than $3\epsilon$, we see that all those arcs $\{\beta_i: \ i\ge i_0\}$ are disjoint. Moreover, we can further infer that no two of them may be contained in the same component of $A\setminus\partial\Omega$, where $A$ denotes the closed annulus with boundary circles $\partial D_r(x)$ and $\partial D_\epsilon(x)$. Indeed, if this happens for $\beta_i, \beta_j$ with $k\ne j\ge i_0$ then $\beta_k\cup\beta_j$ lies in a component $P$ of $A\setminus \partial\Omega$, which is necessarily a subset of $\Omega$. In such a case the union $\gamma_k\cup\beta_k\cup P\cup\beta_j\cup\gamma_j$ would be a connected subset of $\Omega$ that contains $x_k, x_j$ both and is of diameter $<2\epsilon$. This is prohibited, by the choices of $\{x_i\}$.

Therefore, if we denote by $P_i (i\ge i_0)$ the component of $A\setminus \partial\Omega$ that contains $\beta_i$ then $P_i\cap P_j=\emptyset$ for all $i\ne j\ge i_0$, indicating that $A\setminus\partial\Omega$ has infinitely many components that intersect the two circles $\partial D_r(x)$ and $\partial D_\epsilon(x)$ both. By [@LLY-2019 Definition 4], we see that the Schönflies relation on $\partial\Omega$ is not trivial. Thus, by [@LLY-2019 Theorem 7] we can infer that $\partial\Omega$ is not a Peano compactum. This is absurd, since we assume $\partial\Omega$ to be a Peano compactum.

Theory of Cluster Sets for Generalized Jordan Domains {#cluster_set}
-----------------------------------------------------

In this subsection we recall from [@CL66] some elements of cluster sets and characterize generalized Jordan domains as those that are simply connected at the boundary.

For the sake of convenience, we will focus on continuous maps $h$ defined on generalized Jordan domains $U\subset\hat{{\mathbb{C}}}$. Since a Jordan curve separates $\hat{{\mathbb{C}}}$ into two domains, we see that $\partial U$ contains at most countably many components that are Jordan curves. Denote these boundary components of $U$ as $\{\Gamma_n\}$. Moreover, denote by $W_n$ the components of $\hat{{\mathbb{C}}}\setminus\Gamma_n$ that is disjoint from $U$. Here we are mostly interested in the case when $U$ is a circle domain and when $h$ is conformal.

Given a continuous map $h: U\rightarrow V\subset\hat{{\mathbb{C}}}$. The cluster set $C(h,z_0)$ for $z_0\in\partial U$ is defined as $$\bigcap_{r>0}\overline{h(D_r(z_0)\cap U)},$$ where $D_r(z_0)=\{z: |z-z_0|<r\}$. This is a nonempty compact set, since these closures $\overline{h\left(D_r(z_0)\cap U\right)}$ with $r>0$ are considered as subsets of $\hat{{\mathbb{C}}}$. In the following, we will obtain the connectivity of all of them, by showing that [*every neighborhood of an arbitrary point $x\in\partial U$ contains a smaller neighborhood $N_x$ (in $\hat{{\mathbb{C}}}$) with $N_x\cap U$ connected.*]{} A domain with this property will be said to be [**simply connected at the boundary**]{}. This is a special sub-case for the property of being finitely connected at the boundary.

\[connected\_cluster\] Each generalized Jordan domain is simply connected at the boundary. Consequently, if $h: U\rightarrow\hat{{\mathbb{C}}}$ is a continuous map every cluster set $C(h,z_0)$ with $z_0\in\partial U$ is a continuum. In particular, if $h$ is a homeomorphism its cluster sets are sub-continua of $\partial h(U)$.

We need [**Zoretti Theorem**]{} [@Whyburn64 p.35,Corollary 3.11], which reads as follows.

\[Zoretti\] If $K$ is a component of a compact set $M$ (in the plane) and $\epsilon$ is any positive number, then there exists a simple closed curve $J$ which encloses $K$ and is such that $J\cap M=\emptyset$, and every point of $J$ is at a distance less than $\epsilon$ from some point of $K$.

By [**Zoretti Theorem**]{}, We only consider the case that $z_0$ lies on a non-degenerate boundary component $\Gamma_p$ for some $p\ge1$, which is a Jordan curve. By the well known Schönflies Theorem [@Moise p.72,Theorem 4], we may assume that $\Gamma_p=\{|z|=1\}$ and $U\subset {\mathbb{D}}^*:=\{|z|>1\}\subset\hat{{\mathbb{C}}}$.

Given an open subset $V_0$ of $\hat{{\mathbb{C}}}$ that contains $z_0$, we may fix a closed geometric disk $D$ on $\hat{{\mathbb{C}}}$ that is centered at $z_0$ and is such that $(D\cap U)\subset V_0$. Denote by $\rho$ the distance between $D$ and $\hat{{\mathbb{C}}}\setminus V_0$. Since $U$ has property $S$, we may find finitely many regions that are of diameter less than $\rho$, say $M_n(1\le n\le N)$, so that $\bigcup_nM_n=U$ and that every $M_n$ has property $S$. See for instance [@Whyburn42 p.21, Theorem (15.41)].

Let $W$ be the union of all those $M_n$ with $z_0\in\overline{M_n}$. Renaming the regions $M_n$, we may assume that $z_0\in\overline{M_n}$ if and only if $1\le n\le N_0$ for some integer $N_0<N$.

Using [**Zoretti Theorem**]{} repeatedly, we may choose a sequence of Jordan curves $\gamma_k\subset U$ that converge to $\Gamma_p$ under Hausdorff distance. Fix a point $z_k\in\gamma_k$ that is not contained in $D$, so that $z_\infty=\lim\limits_{k\rightarrow\infty}z_k\in\Gamma_p$. Assume that every $\gamma_k$ is parameterized as $g_k: [0,1]\rightarrow U$, with $g_k(0)=g_k(1)=x_k$, so that $g_k(t)$ traverses along $\gamma_k$ counter clockwise as $t$ runs through $[0,1]$.

Fix a point $w_0\in U$ that lies in ${\mathbb{D}}^*\cap\partial D$, an open arc that is separated by $w_0$ into two open arcs, say $a$ and $b$. Going to an appropriate sub-sequence, if necessary, we may assume that every $\gamma_k$ separates $w_0$ from $z_0$ thus intersects both $a$ and $b$. Let $x_k\in\gamma_k$ be the last point at which $\gamma_k$ leaves $a$. Let $y_k\in\gamma_k$ be the first point, after $x_k$, that lies on $b$. Denote by $\alpha_k$ the sub-arc of $\gamma_k$ lying in $D$ that connects $x_k$ to $y_k$. Then $\alpha_k$ converges to the arc $D\cap\Gamma_p$ under Hausdorff distance.

Now, let $W_k$ be the union of all these $M_n(1\le n\le N)$ that intersects $\alpha_k$. Then $W_k$ is connected hence is a region, that contains the whole arc $\alpha_k$. Since there are finitely many choices for the regions $M_n$, we can find an infinite subsequence, say $\{k_i: i\ge1\}$, such that these regions $W_{k_i}$ coincide with each other.

We claim that each of these regions $W_{k_i}$ contains $W$. With this we see that for any open disk $D_r(z_0)\subset D$ with $r$ small enough (say, smaller than the distance from $z_0$ to $U\setminus W$), the union $V_1=W_{k_1}\cup D_r(z_0)$ is an open subset of $\hat{{\mathbb{C}}}$ we are searching for. This $V_1$ contains $z_0$, lies in $V_0$, and is such that $V_1\cap U$ is connected.

To verify the above mentioned claim, we connect $z_0$ to a point $w_n\in M_n$ by an open arc $\beta_n\subset M_n$ for $1\le n\le N_0$. Since $\lim\limits_{k\rightarrow\infty}\alpha_k=D\cap\Gamma_p$ under Hasudorff distance and since $z_0$ is the center of $D$, we see that $\beta_n$ and hence $M_n$ intersects $\alpha_{k_i}$ for infinitely many $i$. From this we can obtain $M_n\subset W_{k_1}$ for $1\le n\le N_0$, indicating that $W\subset W_{k_i}$ for all $k_i$.

In [@Ntalampekos-Younsi19 Proposition 3.5], Ntalampekos and Younsi obtain the result of Theorem \[connected\_cluster\], assuming in addition that $f$ be a homeomorphism of a generalized Jordan domain $D$ onto another planar domain. In Theorem \[connected\_cluster\], we only require that $f$ be a continuous map and the codomain may not be the complex plane or the extended complex plane. Our arguments are more direct and the whole proof is shorter. Moreover, we do not use Moore’s decomposition theorem [@Moore25]; actually we can not refer to this famous theorem, since $h$ may send $D$ into an arbitrary space. We refer to [@Ntalampekos-Younsi19 Theorem 3.6] and [@Ntalampekos-Younsi19 Lemma 3.7] for details concerning the roles that Moore’s decomposition theorem plays in the proof for [@Ntalampekos-Younsi19 Proposition 3.5].

There is another merit of Theorem \[connected\_cluster\] that is noteworthy, if one wants to characterize all planar domains that are simply connected at the boundary. By Theorem \[connected\_cluster\], a generalized Jordan domain is such a region. On the other hand, Theorem \[topology\_metric\] ensures that a region simply connected at the boundary necessarily has property $S$. For such a region $U$, all of its boundary components are Peano continua. Moreover, the assumption of simple connectedness at the boundary implies that none of them has a cut point. This means that the region $U$ is necessarily a generalized Jordan domain.

From this we can infer a nontrivial criterion for generalized Jordan domain, in terms of simple connectedness at the boundary. This provides another justification for the introduction of generalized Jordan domain as a new term.

\[jordan\] A planar domain is simply connected at the boundary if and only if it is a generalized Jordan domain.

A Topological Counterpart for Generalized Continuity Theorem {#outline}
------------------------------------------------------------

This subsection proves [**Theorem \[topological-cct\]**]{}, a topological counterpart for [**Theorems \[arsove\] and \[arsove\_sigma\]**]{}.

To begin with, let us recall a recent result by He and Schramm: [*each countably connected domain $\Omega\subset\hat{{\mathbb{C}}}$ is conformally homeomorphic to a circle domain $D$, unique up to Möbius equivalence*]{} [@He-Schramm93 Theorem 0.1]. Slightly later, they even prove that any domain $\Omega\subset\hat{{\mathbb{C}}}$ is conformally equivalent to some circle domain (1) if $\partial\Omega$ has at most countably many components that are not geometric circles or single points and (2) if the collection of those components has a countable closure in the space formed by all the components of $\partial\Omega$ [@He-Schramm95b; @He-Schramm95a]. However, Koebe’s conjecture is still open if $\partial\Omega$ has a complicated part like a cantor set of segments. Therefore, we may focus on domains $\Omega$ such that the boundary $\partial\Omega$ is “simple” in some sense, say from a topological point of view.

In other words, we would like to limit our discussions to the case when $\partial\Omega$ does not possess a difficult topology. To this end, we examine the necessary conditions for $\varphi: D\rightarrow \Omega$ to have a continuous extension to the closure $\overline{D}$. At this point, we even do not assume the homeomorphism $\varphi:  D\rightarrow \Omega$ to be conformal.

\[necessary\] If a homeomorphism $\varphi:  D\rightarrow \Omega$ of a generalized Jordan domain $D$ admits a continuous extension to $\overline{D}$ then $\partial\Omega$ is a Peano compactum and $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for $z_0\in\partial D$.

Here $\displaystyle\sigma_r(z_0)=\sup_{D\cap C_r(z_0)}|\varphi(z_1)-\varphi(z_2)|$, with $C_r(z_0)=\{z: \ |z-z_0|=r\}$. This quantity is often called the [**oscillation**]{} of $\varphi$ on $C_r(z_0)\cap D$. Clearly, the uniform continuity of $\overline{\varphi}:\ \overline{D}\rightarrow\overline{\Omega}$ indicates that $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial D$. So the only thing to be verified is that the boundary $\partial\Omega$ is a Peano compactum.

Assume that $\varphi$ has a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$. Since $D$ is a generalized Jordan domain, it has Property $S$. Then the uniform continuity of $\overline{\varphi}$ ensures that $\Omega$ also has Property $S$, which then indicates that $\partial\Omega$ is a Peano compactum.

The “only if” part of Theorem \[topological-cct\] is given in Theorem \[necessary\]. Before we continue to prove the “if” part, we want to mention some basic observations that are noteworthy. Firstly, the union of finitely many Peano continua is a Peano compactum. Secondly, if $\varphi$ is conformal then we always have $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ provided that the following are satisfied:

-   $D$ has finitely many boundary components and each of them is locally connected,

-   $\varphi: D\rightarrow\Omega$ is a conformal homeomorphism.

Therefore, Theorem \[topological-cct\] includes a simple case that extends the [**Continuity Theorem**]{} to the case of finitely connected circle domains $D$. Finally, the proof for [@Arsove68-a Theorem 1] already contains the necessary elements that will lead us to the result of Theorem \[topological-cct\], which includes Arsove’s theorem [@Arsove68-a Theorem 1] as a special subcase. In order to provide a self-contained argument and to make concrete clarifications, that become necessary when we involve infinitely connected domains, we also provide a proof for Theorem \[topological-cct\] that comes from a slight modification of Arsove’s proof for [@Arsove68-a Theorem 1]. Exactly the same argument is used in [@Arsove67 Lemma 2] which, as well as that used in [@Arsove68-a Theorem 1], employs the property of being locally sequentially connected. Here we follow the same line of arguments, as those adopted in [@Arsove68-a Theorem 1]. The only difference is that we use [**Property $S$**]{}, instead of the property of [**being locally sequentially accessible**]{}.

Let $\varphi$ be a homeomorphism of a generalized Jordan domain $D$ onto a domain $\Omega\subset\hat{{\mathbb{C}}}$. Suppose that $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial D$ and that $\partial\Omega$ is a Peano compactum. It will suffice if we can show that each cluster set $C(\varphi,z_0)$ is a singleton.

Suppose on the contrary that the cluster set $C(\varphi,z_0)$ at $z_0\in\partial D$ contains two points, say $w_1\ne w_2$. Then we can find an infinite sequence $z_n\rightarrow z_0$ of distinct points satisfying $\varphi\left(z_{2n-1}\right)\rightarrow w_1$ and $\varphi\left(z_{2n}\right)\rightarrow w_2$.

Since $\partial\Omega$ is a Peano compactum, by Theorem \[property\_s\] we see that $\Omega$ has Property $S$. That is to say, for any number $\varepsilon>0$ we can find finitely many connected subsets of $\Omega$, say $N_1,\ldots,N_k$, satisfying $\displaystyle \bigcup_iN_i=\Omega$ and $\displaystyle\max_{1\le i\le k}{\rm diam}(N_i)<\varepsilon$.

Choose a positive number $\varepsilon<\frac13|w_1-w_2|$. Then, there exist two of those connected sets $N_i$, say $N_1$ and $N_2$, such that

-   $N_1$ contains infinitely many points in $\{\varphi\left(z_{2n-1}\right)\}$,

-   $N_2$ contains infinitely many points in $\{\varphi\left(z_{2n}\right)\}$.

Since $z_n\rightarrow z_0$ and since $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$, we can further choose a small enough number $r>0$ such that $\sigma_r(z_0)<\varepsilon$ and that the intersections $\displaystyle N_1\cap\{\varphi\left(z_{2n-1}\right)\}$ and $\displaystyle N_2\cap \{\varphi\left(z_{2n}\right)\}$ each contains at least one point outside $\varphi(D_r(z_0)\cap D)$ and at least one point inside. Therefore, we have $N_i\cap\varphi(C_r(z_0))\ne\emptyset$ for $i=1,2$.

Let $M$ be the union of $\{w_1\}\cup N_1, \ \varphi(C_r(z_0))$, and $\{w_2\}\cup N_2$. As $\sigma_r(z_0)$ is defined to be the diameter of $\varphi(C_r(z_0))$, we have $|w_1-w_2|\le{\rm diam}(M)<3\varepsilon$. This is absurd, since we have chosen $\varepsilon<\frac13|w_1-w_2|$.

A Topological Counterpart for Generalized OTC Theorem {#outline-otc}
-----------------------------------------------------

This subsection proves Theorem \[topological-otc\].

To this end, we firstly investigate into the boundary behaviour of an arbitrary homeomorphism $\varphi: D\rightarrow\Omega$ of a generalized Jordan domain $D$, which has a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ to the whole closure $\overline{D}$. Here we recall that a generalized Jordan domain is a planar domain that satisfies the following two properties:

1.  $\partial U$ is a Peano compactum,

2.  each component of $\partial U$ is either a point or a Jordan curve.

We have the following result, from which the “if” part of Theorem \[topological-otc\] is easily inferred.

\[boundary\_property\] The restriction map $\overline{\varphi}_Q: Q\rightarrow P=\overline{\varphi}(Q)$ to any component $Q$ of $\partial D$ is non-alternating. Moreover, the whole extension $\overline{\varphi}: \overline{D}\rightarrow \overline{\Omega}$ is a monotone map if and only if $\Omega$ is also a generalized Jordan domain.

Under the assumption in Theorem \[boundary\_property\], the boundary $\partial\Omega$ is a Peano compactum. Therefore, by Torhorst Theorem [@Kuratowski68 p.512, $\S61$, II, Theorem 4], we can infer that $\Omega$ is a generalized Jordan domain if and only if no component of its boundary $\partial\Omega$ has a cut point. Therefore, the “only if” part is indicated by Theorem \[necessary\]. Together with the above Theorem \[boundary\_property\], we have provided a complete proof for Theorem \[topological-otc\].

We firstly obtain the first half of the above theorem, showing that $\left.\overline{\varphi}\right|_{Q}$ is non-alternating for any component $Q$ of $\partial D$.

Recall that a continuous map $f: A\rightarrow B$ is called a non-alternating transformation provided that for no two points $x,y\in B$ does there exist a separation $A\setminus f^{-1}(x)=A_1\cup A_2$ such that $y$ lies in $f(A_1)\cap f(A_2)$ [@Whyburn42 p.127, (4.2)]. From this one can infer that $f:A\rightarrow B$ is non-alternating if and only if $f(A_1)\cap f(A_2)=\emptyset$ for any $x\in B$ and for any separation $A\setminus f^{-1}(x)=A_1\cup A_2$.

By [**Zoretti Theorem**]{}, the image $P=\overline{\varphi}(Q)$ is a component of $\partial\Omega$. By definition of non-alternating transformation, we only need to show that $\overline{\varphi}(A_1)\cap \overline{\varphi}(A_2)=\emptyset$ for any $x\in P$ and for any separation $Q\setminus\left(\overline{\varphi}\right)^{-1}\!(x)=A_1\cup A_2$.

Assume on the contrary that there were a point $x\in P$ and a separation $Q\setminus\left(\overline{\varphi}\right)^{-1}\!(x)=A_1\cup A_2$ such that $\overline{\varphi}(z_1)=\overline{\varphi}(z_2)$ for $z_i\in A_i (i=1,2)$. Set $x'=\overline{\varphi}(z_1)=\overline{\varphi}(z_2)$. Since $D$ is a generalized Jordan domain, the component $Q$ of $\partial D$ must be a simple closed curve. Thus the point inverse $\left(\overline{\varphi}\right)^{-1}\!(x)$ contains two points $y_1\ne y_2$ such that $\{y_1,y_2\}$ separates $z_1$ from $z_2$ in $Q$. Since $D$ has property $S$, all boundary points of $D$ are accessible from $D$. Thus we can find an open arc $\alpha\subset D$ that connects $y_1$ to $y_2$. From this we see that $Q\cup\alpha$ is a $\theta$-curve and that $D\setminus\alpha$ consists of two domains. Let $U_i (i=1,2)$ be the one whose boundary contains $z_i$. Clearly, $J=\varphi(\alpha)\cup\{x\}$ is a Jordan curve and $\Omega\setminus J=\varphi(U_1)\cup \varphi(U_2)$. See the left part of Figure \[5.1\],

-0.25cm

[cccc]{}

(0,0) circle (5); (3,4) – (11,4) –(11,-4)– (3,-4); at (8,-4.5) [$\alpha$]{};

(3,4)circle (0.2); at (2.2,3.7) [$y_1$]{}; (3,-4)circle (0.2); at (2.2,-3.7) [$y_2$]{}; (5,0)circle (0.2); at (4.0,0) [$z_2$]{}; (-5,-0)circle (0.2); at (-4.0,0) [$z_1$]{};

at (8,5) [$U_1$]{}; at (8,0) [$U_2$]{};

&&&

(0,0) circle (5); (3,4) – (11,4) –(11,-4)– (3,-4); at (8,-4.5) [$\alpha$]{};

(-5,0) – (-5,5.4) – (8,5.4)– (8,4); (8,4)– (8,2)– (14,2) – (14,0)–(11,0); (11,0)–(5,0); at (15,1) [$\beta$]{};

(-5,0) – (-5,5.4) – (8,5.4)– (8,4); at (2,6.2) [$\beta_1$]{}; (8,4) – (11,4) –(11,0);

(8,4)–(11,4)–(11,0); at (12,4) [$\beta_3$]{}; (11,0)–(5,0); at (8,-1) [$\beta_2$]{};

(3,4)circle (0.2); at (2.2,3.7) [$y_1$]{}; (3,-4)circle (0.2); at (2.2,-3.7) [$y_2$]{}; (5,0)circle (0.2); at (4.0,0) [$z_2$]{}; (-5,-0)circle (0.2); at (-4.0,0) [$z_1$]{};

(8,4)circle (0.2); at (8.7,4.7) [$b_1$]{}; (11,0)circle (0.2); at (11.7,-0.7) [$b_2$]{};

-0.25cm

for relative locations of the arc $\alpha$, the domains $U_i$ and the points $y_i,z_i$. Now, fix an arc $\beta\subset D$ that connects $z_1$ to $z_2$ and denote by $\beta_i\subset(U_i\cap\beta)$ the maximal open sub-arc of $\beta$ that has $z_i$ as one of its ends. Denote by $b_i$ the other end point of $\beta_i$ for $i=1,2$. Obviously, we have $b_1,b_2\in\alpha$. Let $\beta_3$ be the closed sub-arc of $\alpha$ with ends $b_1,b_2$. Then we have an arc $\beta'=\beta_1\cup\beta_2\cup\beta_3$, lying in $D$ and intersecting $\alpha$ at $\beta_3$. See right part of Figure \[5.1\].

Since $\varphi: D\rightarrow\Omega$ is a homeomorphism, we know that $\varphi(\beta')=\varphi(\beta_1)\cup\varphi(\beta_2)\cup\varphi(\beta_3)$ is an arc contained in $\Omega$ such that (1) $\varphi(\beta')\cap\varphi(\alpha)=\varphi(\beta_3)$ and (2) $\varphi(\beta_i)\subset \varphi(U_i)$ for $i=1,2$. Since the simple closed curve $J=\{x\}\cup\varphi(\alpha)$ does not contain the point $x'=\overline{\varphi}(z_1)=\overline{\varphi}(z_2)$ and since each of $\varphi(\beta_i)$ has $x'$ as one of its ends, we can infer that $\varphi(\beta_1)$ and $\varphi(\beta_2)$ are both contained in a single component of $\hat{{\mathbb{C}}}\setminus J$, thus are both contained in a single component of $\Omega\setminus J$, which is either $\varphi(U_1)$ or $\varphi(U_2)$. This is absurd, since we have chosen $\beta_i\subset U_i(i=1,2)$ so that $\varphi(\beta_i)\subset\varphi(U_i)$.

Then we go on to consider the latter half of Theorem \[boundary\_property\]. Since the“only if" part of which is obvious, we just discuss the “if” part. To this end, we recall that a special type of non-alternating maps come from the family of [*monotone maps*]{}. If we confine ourselves to continuous maps between compacta then, under a monotone map $f: X\rightarrow Y$, the pre-image of any point $y\subset Y$ is a sub-continuum of $X$. Therefore, if $P$ is a component of $\partial\Omega$ with $\varphi^B(Q)=P$ and if $P$ is a single point or is a Jordan curve then it has no cut point and hence the inverse $\overline{\varphi}^{-1}(x)$ for any $x\in P$ is a sub-continuum of $Q$. This means that the restriction $\left.\overline{\varphi}\right|_Q$ is monotone. Therefore, the whole extension $\overline{\varphi}$ is monotone provided that $\Omega$ is a generalized Jordan domain, too.

On Domains $\Omega\subset\hat{{\mathbb{C}}}$ Whose Boundary is a Peano Compactum  {#topology}
=================================================================================

In this section we will provide a complete proof for Theorem \[topology\_metric\]. Namely, we shall prove that the following six conditions are equivalent for all domains $\Omega\subset\hat{{\mathbb{C}}}$:

1.  $\partial\Omega$ is a Peano compactum.

2.  $\Omega$ has property S.

3.  All points of $\partial\Omega$ are locally accessible.

4.  All points of $\partial\Omega$ are locally sequentially accessible.

5.  $\Omega$ is finitely connected at the boundary.

6.  The completion $\overline{\Omega}_d$ of the metric space $(\Omega,d)$ is compact.

Our arguments will center around two groups of implications: $(1)\Leftrightarrow(2)\Leftrightarrow(5)\Leftrightarrow(6)$ and $(2)\Rightarrow(3)\Rightarrow(1)\Rightarrow(4)\Rightarrow(1)$. The equivalence $(5)\Leftrightarrow(6)$ has been given in [@BBS16 Theorem 1.1]. The equivalence $(1)\Leftrightarrow(2)$ is obtained by Theorem \[property\_s\] in the previous section. The equivalence $(1)\Leftrightarrow(5)$ is to be established in Theorem \[S\_finitely\_connected\]. The implication $(2)\Rightarrow(3)$ is already known [@Whyburn42 p.111, (a)] and the implications $(3)\Rightarrow(1)\Rightarrow(4)\Rightarrow(1)$ will be discussed in Theorem \[3-1-4-1\].

There are three issues we want to mention. Firstly, the notion of [**local accessibility**]{} coincides with that of [**regular accessibility**]{} in [@Whyburn42 p.112, Theorem (4.2)]. Here a point $x\in\partial\Omega$ is locally accessible from $\Omega$ if for any $\epsilon>0$ there is a number $\delta>0$ such that all points $z\in\Omega$ with $|z-x|<\delta$ may be connected to $x$ by a simple arc inside $\Omega\cup\{x\}$, whose diameter is smaller than $\epsilon$. Secondly, a point $\xi\in\partial\Omega$ is called [**locally sequentially accessible**]{} if for each $r>0$ and for each sequence $\{\xi_n\}$ of points in $\Omega$ that converge to $\xi$ the common part $\Omega\cap D_r(\xi)$, of $\Omega$ and the open disk $D_r(\xi)$ centered at $\xi$ with radius $r$, is an open set such that one of its components contains infinitely many $\xi_n$. Lastly, a domain $\Omega\subset\hat{{\mathbb{C}}}$ is [**finitely connected at the boundary point $x\in\partial\Omega$**]{} provided that for any number $r>0$ there is an open subset $U_x$ of $\hat{{\mathbb{C}}}$, lying in $D_r(x)$, such that $U_x\cap\Omega$ has finitely many components. In particular, if we further require that $U_x\cap\Omega$ be connected, we say that $\Omega$ is [**simply connected at $x$**]{}. If $\Omega$ is finitely connected at every of its boundary points, we say that $\Omega$ is [**finitely connected at the boundary**]{}. Similarly, if $\Omega$ is simply connected at every of its boundary points, we say that $\Omega$ is [**simply connected at the boundary**]{}. See Theorem \[jordan\] for a nontrivial characterization generalized Jordan domain, as planar domains that are simply connected at the boundary.

\[S\_finitely\_connected\] $\Omega$ has property S if and only if it is finitely connected at the boundary.

Suppose that $\Omega$ is finitely connected at the boundary. Given an arbitrary number $r>0$, we can find for any $x\in\partial\Omega$ an open set $G_x\subset\{z: |z-x|<\frac{r}{2}\}$ such that $G_x\cap\Omega$ has finitely many components [@BBS16 Definition 2.2]. Clearly, the collection $\{G_x: x\in\partial\Omega\}$ gives an open cover of the boundary $\partial\Omega$. So we can find a finite sub-cover of $\partial \Omega$, denoted as $\{G_1,\ldots, G_n\}$. Since $\Omega\setminus\left(\bigcup G_i\right)$ is a compact subset of $\Omega$, we can cover it with finitely many small disks contained in $\Omega$, with radius $<\frac{r}{2}$. For $1\le i\le n$ the intersection $G_i\cap\Omega$ has finitely many components. These components and the above-mentioned small disks, that cover $\Omega\setminus\left(\bigcup G_i\right)$, form a finite cover of $\Omega$ by sub-domains of $\Omega$ having a diameter $<r$. This shows that $\Omega$ has property S.

On the other hand, assuming that $\Omega$ has property S. Given an arbitrary point $x\in\partial\Omega$ and any positive number $r$, we can cover $\Omega$ by finitely many domains $W_1,\ldots, W_N\subset\Omega$ of arbitrarily small diameter, say $\varepsilon\in(0,\frac{r}{3})$. Denote by $U_x$ the union of all those $W_i$ whose closure contains $x$ and by $E_x$ the union of all those $W_i$ whose closure does not contain $x$. Then $\overline{E_x}$ is a compact set, whose distance to $x$ is a positive number $r_x>0$. Let $$G_x=U_x\cup\{x\}\cup\left\{z\notin\Omega: |z-x|<\min\left\{\frac{r}{3},r_x\right\}\right\}.$$ Then $G_x\subset\left\{z: |z-x|<\frac{r}{2}\right\}$ is an open set with $G_x\cap\Omega=U_x$, which is the union of some of the domains $W_1,\ldots, W_N$ and hence has finitely many components. This verifies that $\Omega$ is finitely connected at $x$. Since $x$ and $r>0$ are both flexible we see that $\Omega$ is finitely connected at the whole boundary.

\[3-1-4-1\] The implications $(3)\Rightarrow(1)\Rightarrow(4)\Rightarrow(1)$ hold. Thus Theorem \[topology\_metric\] is true.

Without losing generality, we may assume that $\infty\in\Omega$. Under this context $\partial\Omega$ may be considered as a compactum on ${\mathbb{C}}$.

Let us start from the implications $(3)\Rightarrow(1)$ and $(4)\Rightarrow(1)$, which will be obtained by a contrapositive proof.

Suppose on the contrary that $\partial\Omega$ were not a Peano compactum. Then it would not satisfy the Schönflies condition [@LLY-2019 Theorem 3]. In other words, there would exist an unbounded closed strip $W$, whose boundary consists of two parallel lines $L_1\ne L_2$, such that $W\cap\partial\Omega$ has infinitely many components, say $W_n$ for $n\ge1$, each of which intersects both $L_1$ and $L_2$. See for instance [@LLY-2019 Lemma 3.8]. Let $L$ be the line parallel to $L_1$ with $${\rm dist}(L,L_1)={\rm dist}(L,L_2).$$ Then $L$ intersects $W_n$ for all $n\ge1$. Pick an infinite sequence of points $z_n\in (W_n\cap L)$ which converge to a limit point $z_0\in\partial\Omega$. Pick a point $\xi_n\in \Omega$ such that $\lim\limits_{n\rightarrow\infty}|\xi_n-z_n|=0$.

Clearly, for infinitely many choices of $n\ge1$, no arc connecting $\xi_n$ to $z_0$ is disjoint from $L_1\cup L_2$. Thus $z_0$ is not locally accessible from $\Omega$. This verifies the implication $(3)\Rightarrow(1)$. On the other hand, if we fix a neighborhood $V_0$ of $z_0$, which entirely lies in the interior of $W$, then there are infinitely many $\xi_n$ that belong to distinct components of $V_0\cap \Omega$. This indicates that $z_0$ is not locally sequentially accessible from $\Omega$ and verifies the implication $(4)\Rightarrow(1)$.

The rest of our proof is to verify the implication $(1)\Rightarrow(4)$. And we will follow the ideas used in the proof for [@Arsove67 Lemma 1]. Indeed, if we suppose on the contrary that some point $z_0\in\partial\Omega$ were not locally sequentially accessible from $\Omega$, then for some $\rho>0$ there would exist infinitely many components of $\Omega\cap D_\rho(z_0)$, with $D_\rho(z_0)=\{z: |z-z_0|\le\rho\}$, that intersect the smaller disk $D_{\rho/2}(z_0)$. Denote these components by $Q_n (n\ge1)$. Since each $Q_n$ intersects $C_\rho(z_0)=\{z: |z-z_0|=\rho\}$ and since each of them is path connected, we can find paths $\gamma_n\subset Q_n$, lying in $A_\rho(z_0)=\{z: \frac{\rho}{2}\le |z-z_0|\le\rho\}$, that connects a point on $C_\rho(z_0)$ to a point on $C_{\rho/2}(z_0)$. Let $P_n$ be the component of $Q_n\cap A_{\rho}(z_0)$ that contains $\gamma_n$. Clearly, all these $P_n(n\ge1)$ are each a component of $\Omega\cap A_\rho(z_0)$. From this we may conclude that the Schönflies relation $R_{\partial\Omega}$ contains a pair $(z_1,z_2)$ for some $z_1\in C_\rho(z_0)$ and some $z_2\in C_{\rho/2}(z_0)$. See [@LLY-2019 Lemma 3.8] and [@LLY-2019 Remark 3.9] for this conclusion. Thus $\partial\Omega$ is not a Peano compactum, since a compact $K\subset\hat{{\mathbb{C}}}$ is a Peano compactum if and only if $R_K$ is a trivial relation.

To Generalize Continuity Theorem [— the first approach]{} {#outline-1}
=========================================================

Our target of this section is to give a complete proof for Theorem \[arsove\].

Since Theorem \[topological-cct\] provides the “only if” part, we just discuss the “if” part. And the only problem is that, for domains $\Omega\subset\hat{{\mathbb{C}}}$ whose boundary $\partial\Omega$ is a Peano compactum having countably many non-degenerate components $\{P_n\}_{n=1}^\infty$, it is not known whether $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ holds for all $z_0\in\partial D$. We will obtain the following special case for Theorem \[arsove\].

\[sufficient\] Given a circle domain $D$ and a conformal homeomorphism $\varphi: D\rightarrow \Omega$, where the boundary $\partial\Omega$ has countably many non-degenerate components $\{P_n\}$ with $\displaystyle\sum_n{\rm diam}(P_n)<\infty$ and all its point components form a set of [**zero**]{} linear measure. If $\partial\Omega$ is a Peano compactum then $\varphi$ has a continuous extension to $\overline{D}$.

Theorem \[sufficient\] is benefited from ideas used in the main theorem of [@Arsove67], which reads as follows.

Each of the following is necessary and sufficient for a bounded simply connected plane region $\Omega$ to have its boundary parametrizable as a closed curve [**(equivalently, being a Peano continuum)**]{}:

1.  all points of $\partial\Omega$ are locally accessible,

2.  all points of $\partial\Omega$ are locally sequentially accessible,

3.  some (equivalently, any) Riemann mapping function $\varphi:{\mathbb{D}}\rightarrow \Omega$ for $\Omega$ can be extended to a continuous mapping of $\overline{{\mathbb{D}}}$ onto $\overline{\Omega}$.

Here we use Property $S$ instead of the property of being locally sequentially accessible. As in earlier works, such as [@Arsove67; @Arsove68-a], we also need to estimate from above the oscillations of the homeomorphism $\varphi: D\rightarrow \Omega$. To do that, we assume in addition some control on the diameters of the non-degenerate components of $\partial\Omega$. On the other hand, we also need to deal with the point components of $\partial\Omega$, by assuming that they form a set that is small in terms of linear measure.

In order to prove Theorem \[sufficient\], we only need to obtain the following Theorem \[oscillation\].

Our proof for Theorem \[oscillation\] uses a bijection between the boundary components of $D$ and those of $\Omega$. This bijection associates to any component $Q$ of $\partial D$ a component $P$ of $\partial\Omega$, which actually consists of all the cluster sets $C(\varphi,z_0)$ with $z_0\in Q$. In deed, by [**Zoretti Theorem**]{}, we can choose inductively an infinite sequence of simple closed curves $\Gamma_n\subset D$ such that for all $n\ge1$ we have: (1) every point of $\Gamma_n$ is at a distance less than $\frac{1}{n}$ from a point of $Q$; and (2) $\Gamma_{n+1}$ separates $Q$ from $\Gamma_n$. Let $U_n$ be the component of $\hat{{\mathbb{C}}}\setminus\varphi(\Gamma_n)$ that contains $\varphi(\Gamma_{n+1})$. Then $\{U_n\}$ is a decreasing sequence of Jordan domains with $\overline{U_{n+1}}\subset U_n$ for all $n\ge1$. Therefore, we know that $M=\cap_nU_n=\cap_n\overline{U_n}$ is a sub-continuum of $\hat{{\mathbb{C}}}\setminus U$, whose complement is connected. Consequently, $P=\partial M$ is a sub-continuum of $\partial\Omega$ and is a component of $\partial\Omega$, which consists of all the cluster sets $C(\varphi,z_0)$ with $z_0\in D$.

Following He and Schramm [@He-Schramm93], we set $\varphi^B(Q)=P$. This gives a well defined bijection between boundary components of $D$ and those of $\Omega$. We can infer Theorem \[sufficient\] by combining Theorem \[topological-cct\] and the theorem below, in which we do not require that $\partial\Omega$ be a Peano compactum. The only assumptions are about the diameters of $P_n$ and about the linear measure of the difference $\partial\Omega\setminus\left(\bigcup_nP_n\right)$, the set consisting of all the point components of $\partial\Omega$. Therefore, the result we obtain here is just the oscillation convergence $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial Q_n$, without mentioning the cluster sets $C(\varphi,z_0)$ for $z_0\in\partial Q_n$.

\[oscillation\] Given a circle domain $D$ and a conformal homeomorphism $\varphi: D\rightarrow \Omega$, where the boundary $\partial\Omega$ has countably many non-degenerate components $\{P_n\}$ and all its point components form a set of zero linear measure. Let $Q_n$ be the component of $\partial D$ with $\varphi^B(Q_n)=P_n$ for all $n\ge1$. If there exists an open set $U_n\supset P_n$ satisfying $\displaystyle\sum_{P_k\subset U_n}{\rm diam}(P_k)<\infty$ we have $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial Q_n$.

Let $\Lambda_r(z_0)$ be the [**arc length**]{} of $\varphi(C_r(z_0)\cap D)$, with $C_r(z_0)=\{|z-z_0|=r\}$. Then we have $\displaystyle\inf\limits_{\rho<r<\sqrt{\rho}}\Lambda_r(z_0)\le\frac{2\pi R}{\sqrt{\log1/\rho}}$ for $0<\rho<1$. This result is often referred to as Wolff’s Lemma. See [@Pom92 p.20, Proposition 2.2] for instance. Therefore, $\liminf\limits_{r\rightarrow 0}\Lambda_r(z_0)=0$. This is however different from what we need to verify, which is $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$; since the oscillation $\sigma_r(z_0)$ is defined to be the [**diameter**]{} of $\varphi(C_r(z_0)\cap D)$.

Let $\{k_i: i\ge1\}$ be the collection of all those integers $k_i$ with $P_{k_i}\subset U_n$, arranged so that $k_1<k_2<\cdots$. Recall that $Q_{k_i}$ denotes the component of $\partial D$ with $P_{k_i}=\varphi^B\left(Q_{k_i}\right)$.

Given a point $z_0\in\partial Q_n$ and an arbitrary number $\epsilon>0$, we shall find a positive number $r<\epsilon$ such that $\sigma_r(z_0)<\epsilon$, which then completes our proof.

To this end, we firstly fix a point $w_0\in \Omega$ and then use [**Zoretti Theorem**]{} to find a simple closed curve $\Gamma_i$ for each $P_{k_i}$ such that [**$\Gamma_i$ separates $w_0$ from $P_{k_i}$**]{} and that [**every point of $\Gamma_{k_i}$ is at a distance less than $2^{-i}\epsilon$ from some point of $P_{k_i}$**]{}. Clearly, we have $\displaystyle\sum_{i}{\rm diam}(\Gamma_i)<\infty$. For $i\ge1$, let $W_i^*$ denote the component of $\hat{{\mathbb{C}}}\setminus \Gamma_i$ that contains $P_{k_i}$; moreover, let $W_i$ denote the component of $\hat{{\mathbb{C}}}\setminus \varphi^{-1}(\Gamma_i)$ that contains $Q_{k_i}$.

Then, fixing an integer $N\ge1$ with $\displaystyle \sum_{i=N+1}^\infty{\rm diam}(\Gamma_i)<\frac12\epsilon$, we continue to choose $r>0$ small enough, with $\varphi\left(C_r(z_0)\cap D\right)\subset U_n$, such that $C_r(z_0)\setminus Q_n$ intersects none of the boundary components $Q_{k_1},\ldots, Q_{k_N}$ of $D$. By Wollf’s Lemma, we have $\liminf\limits_{r\rightarrow 0}\Lambda_r(z_0)=0$. Thus we may further require that the above number $r$ is chosen so that $\Lambda_r(z_0)<\frac14\epsilon$.

\[transboundary\_bridge\] Let $F_r$ consist of all the points $q$ in $C_{r}(z_0)\cap \partial D$ such that $\{q\}$ is a component of $\partial D$. Let $F_r^*$ consist of all the points $q^*\in\partial\Omega$ such that $\{q^*\}=\varphi^B(Q)$ for some component $Q$ of $\partial D$ that intersects $C_{r}(z_0)\cap \partial D$. Then the linear measure of $F_r^*$ is zero. Therefore, for the above $\epsilon>0$, we can find a countable cover of $F_r^*$ by open sets of diameter smaller than any constant $\delta>0$, say $\{V_k^*: k\ge1\}$, such that $\sum_j{\rm diam}\left(V_k^*\right)<\frac14\epsilon$.

Since $\partial\Omega$ has at most countably many non-degenerate components and since its point components form a set of zero linear measure, the result of this lemma is immediate.

Now, by flexibility of $\epsilon>0$, we see that the following lemma completes our proof.

\[estimate\_of\_oscillation\] For the above mentioned $r$, the inequality $|\varphi(z_1)-\varphi(z_2)|<\epsilon$ holds for any fixed points $z_1\ne z_2$ lying on $C_{r}(z_0)\cap D$.

To prove this lemma, we may consider the closed sub-arc of $C_{r}(z_0)\setminus Q_n$ from $z_1$ to $z_2$. Denote this arc as $\alpha$. Clearly, it is a compact set disjoint from each of $Q_n, Q_{k_1},\ldots, Q_{k_N}$. Moreover, denote by $M_\alpha$ the union of $\varphi(\alpha\cap D)$ with all the boundary components $\varphi^B(Q)$ of $\Omega$ with $Q$ running through the boundary components of $D$ that intersect $\alpha$. Then, we only need to verify that the diameter of $M_\alpha$ is less than $\epsilon$.

Let us now consider the components $Q$ of $\partial D$, with $Q\cap\alpha\ne\emptyset$, such that $\varphi^B(Q)\subset U_n$ is a non-degenerate component of $\partial\Omega$. These components may be denoted as $Q_j$ for $j$ belonging to an index set ${\mathcal{J}}\subset\{k_1<k_2<\cdots\}$. Clearly, we have ${\mathcal{J}}\subset\{k_i: i\ge N+1\}$.

Let $\left\{V_k^*: k\in{\mathcal{K}}\right\}$ be the cover of $F_r^*$ given in Lemma \[transboundary\_bridge\], so that $\sum_k{\rm diam}(V_k^*)<\frac14\epsilon$. Since all these sets $V_k^*$ are open in $\hat{{\mathbb{C}}}$, we can choose for each point $w\in F_r^*$ a Jordan curve $J_{w}\subset \Omega$ that lies in some $V_k^*$ and separates $w_0$ from the point component $\{w\}$ of $\partial\Omega$. Let $V_w^*$ be the component of $\hat{{\mathbb{C}}}\setminus J_w$ that contains $w$. Let $V_w$ be the component of $\hat{{\mathbb{C}}}\setminus\varphi^{-1}(J_w)$ that contains $\left(\varphi^B\right)^{-1}(\{w\})$, which is the component of $\partial D$ corresponding to $\{w\}$ under $\varphi^B$.

On the other hand, the components of $\alpha\cap D$ form a countable family $\{\alpha_t: t\in{\mathcal{I}}\}$. All these $\alpha_t$ are open arcs or semi-closed arcs on the circle $C_r(z_0)$. In deed, exactly two of them are semi-closed. Now it is easy to see that $$\left\{W_i: i\in{\mathcal{J}}\right\}\ \bigcup\
\left\{V_w: w\in F_m\right\}\ \bigcup\
\left\{\alpha_t: t\in{\mathcal{I}}\right\}$$ is a cover of $\alpha$. Since each $\alpha_t$ is open in $\alpha$, we may choose finite index sets ${\mathcal{J}}_0\subset{\mathcal{J}}$, $F_0\subset F_m$ and ${\mathcal{I}}_0\subset{\mathcal{I}}$, such that $$\left\{W_i: i\in{\mathcal{J}}_0\right\}\ \bigcup\
\left\{V_w: w\in F_0\right\}\ \bigcup\
\left\{\alpha_t: t\in{\mathcal{I}}_0\right\}$$ is a finite cover of $\alpha$. This indicates that $$\left\{W_i^*: i\in{\mathcal{J}}_0\right\}\ \bigcup\
\left\{V_w^*: w\in F_0\right\}\ \bigcup\
\left\{\varphi(\alpha_t): t\in{\mathcal{I}}_0\right\}$$ is a finite cover of $M_\alpha$. Therefore, we can choose a finite subset ${\mathcal{K}}_0\subset{\mathbb{Z}}$ such that $$\left\{W_i^*: i\in{\mathcal{J}}_0\right\}\ \bigcup\
\left\{V_k^*: k\in {\mathcal{K}}_0\right\}\ \bigcup\
\left\{\varphi(\alpha_t): t\in{\mathcal{I}}_0\right\}$$ is a finite cover of $M_\alpha$, too. From this we can infer that, for the above mentioned points $z_1\ne z_2$ lying on $C_r(z_0)\cap D$, the inequality $$|\varphi(z_1)-\varphi(z_2)|<\sum_{j\in{\mathcal{J}}_0}{\rm diam}(\Gamma_j)+\sum_{k}{\rm diam}(V_k^*)+\sum_{t\in{\mathcal{I}}_0}{\rm diam}(\varphi(\alpha_t))<\frac12\epsilon+\frac14\epsilon+\frac14\epsilon=\epsilon$$ always holds. By flexibility of $z_1,z_2\in \alpha\cap D$, this leads to the result of Lemma \[estimate\_of\_oscillation\].

Now we have all the ingredients to construct a proof for Theorem \[arsove\]. To do that, we only need to obtain the result given in Theorem \[oscillation\] under a weaker assumption, saying that the point components of $\partial\Omega$ forms a set of $\sigma$-finite linear measure. Note that, in Theorem \[oscillation\], this set is assumed to be of zero linear measure.

\[oscillation-a\] Given a circle domain $D$ and a conformal homeomorphism $\varphi: D\rightarrow \Omega$, where the boundary $\partial\Omega$ has countably many non-degenerate components $\{P_n\}$ and all its point components form a set of $\sigma$-finite linear measure. Let $Q_n$ be the component of $\partial D$ with $\varphi^B(Q_n)=P_n$ for all $n\ge1$. If there exists an open set $U_n\supset P_n$ satisfying $\displaystyle\sum_{P_k\subset U_n}{\rm diam}(P_k)<\infty$ we have $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial Q_n$.

We shall follow the same ideas in proving Theorem \[oscillation\], except for a couple of minor adjustments. The first one is to infer a slightly more general version of Wolff’s lemma [@Pom92 p.20, Proposition 2.2].

\[Wolff\_lemma\_2\] Let $\varphi$ map a domain $D\subset{\mathbb{C}}$ conformally into a bounded domain $D_R(0)$. Let $C_r(z_0)=\{|z-z_0|=r\}$ and $\Lambda_r(z_0)$ the arc length of $\varphi(C_r(z_0)\cap D)$. Then for any $\epsilon>0$ and any number $\rho\in(0,1)$, there exists $N>0$ such that for all $n>N$, the interval $[\rho^{2^{n+1}},\rho^{2^{n}}]$ has a subset $E_n$ with positive measure such that $\displaystyle\sup_{r\in E_n}\Lambda_r(z_0)<\frac14\epsilon$.

Denote $l(r)=\Lambda_r(z_0)$. Suppose on the contrary that there exists $\epsilon_0>0$ and an increasing sequence $\{n_k: k\ge1\}$ of integers such that $l(r){\geqslant}\epsilon_0$ for almost all $r\in A_n=[\rho^{2^{n+1}},\rho^{2^{n}}]$. Then a simple calculation would lead us to the following inequality $$\int_{A_{n_k}}l^2(r)\frac{dr}{r}{\geqslant}\epsilon^2_0\int_{A_{n_k}}\frac{dr}{r}=
\epsilon^2_0\log\frac{1}{\rho^{2^{n_k}}}$$ for all $k\ge1$. Thus we have $$\displaystyle \int_0^\infty l^2(r)\frac{dr}{r}{\geqslant}\sum_k\int_{A_{n_k}}l^2(r)\frac{dr}{r}=\infty.$$ This is impossible, since Wolff’s lemma states that $\displaystyle \int_0^\infty l^2(r)\frac{dr}{r}{\leqslant}2\pi^2 R^2$. Therefore, the Generalized Wolff’s Lemma holds.

The second adjustment is needed when we prove the result of Lemma \[transboundary\_bridge\]. The aim here is to obtain a number $r$ in the set $E_n$, as defined in the above Lemma \[Wolff\_lemma\_2\], such that $F_r^*$ has zero linear measure. Here we only assume that the point components of $\partial\Omega$ form a set of $\sigma$-finite linear measure.

\[zero linear measure\] Let $F_r, F_r^*$ be defined as in Lemma \[transboundary\_bridge\]. The linear measure of $F_r^*$ is zero for all but countably many of $r\in E_n$.

In this lemma, we only need to consider the case that the point components of $\partial\Omega$ form a set of finite linear measure. Since $\{F_r^*: r\in E_n\}$ are essentially pairwise disjoint Borel sets, in the sense that every two of them has at most countably many common points, one can directly infer the result of Lemma \[zero linear measure\].

Now, we can copy the result and the proof for Lemma \[estimate\_of\_oscillation\], and then infer Theorem \[oscillation-a\]. Combining this with Theorem \[topological-cct\], we readily have Theorem \[arsove\].

The result of Theorem \[oscillation\] still holds, if $D$ is only required to be a generalized Jordan domain. Actually, if $U_0$ denotes the component of $\hat{{\mathbb{C}}}\setminus Q_n$ containing $D$ then we can find a homeomorphism $H:\hat{{\mathbb{C}}}\rightarrow \hat{{\mathbb{C}}}$, sending $Q_n$ onto the unit circle, such that $\left.H\right|_{U_0}$ is conformal map between $U_0$ and $\{ z\in\hat{{\mathbb{C}}}: |z|>1 \}$. In such a way, we see that all the arguments in the proof for Theorem \[oscillation\] still work.

Similarly, all the arguments in the proof for Theorem \[oscillation-a\] are valid, even if the circle domain $D$ is changed into a generalized Jordan domain. Combining this observation with Theorem \[topological-cct\], we can further extend the result of Theorem \[sufficient\] and obtain the following.

\[arsove-new\] Let $\Omega_1$ be a generalized Jordan domain. Let $\varphi: \Omega_1\rightarrow \Omega_2$ be a conformal homeomorphism, where the boundary $\partial\Omega_2$ has at most countably many non-degenerate components $\{P_n\}$ with $\displaystyle\sum_n{\rm diam}(P_n)<\infty$ and all its point components form a set of $\sigma$-finite linear measure. Then $\varphi$ extends continuously to the closure $\overline{\Omega_1}$ if and only if $\partial\Omega_2$ is a Peano compactum.

To Generalize Continuity Theorem [— the second approach]{} {#outline-2}
==========================================================

Our target of this section is to prove Theorem \[arsove\_sigma\].

Let us start from a result that can be inferred as a direct corollary of [@He-Schramm94 Lemma 1.3], which reads as follows.

\[HS-1994\_1.3\] Let $Z\subset{\mathbb{R}}^2$ be a Borel set of $\sigma$-finite linear measure, and let $X\subset{\mathbb{R}}$ be the set of points $x$ such that the section $\left(\{x\}\times{\mathbb{R}}\right)\cap Z$ is uncountable. Then $X$ has zero Lebesgue measure.

In the above lemma, we may consider ${\mathbb{R}}^2$ as the complex plane ${\mathbb{C}}$, consisting of $re^{{\bf i}\theta}$ with $r>0$ and $0\le\theta<2\pi$. Then, we study the set $R_0$ of numbers $r>0$ such that the circle $\left\{re^{{\bf i}\theta}: 0\le\theta<2\pi\right\}$ intersects $Z$ at uncountably many points. For any $r_2>r_1>0$, we see that the part of $Z$ in the annulus $\{z\in{\mathbb{C}}: r_1\le|z|\le r_2\}$ is sent onto the rectangle $[r_1,r_2]\times[0,2\pi]$ by the map $re^{{\bf i}\theta} \mapsto (r,\theta)$. If we define the distance between $r_1e^{{\bf i}\theta_1}$ and $r_2e^{{\bf i}\theta_2}$ to be $|r_1-r_2|+|\theta_1-\theta_2|$, the previous map is actually bi-Lipschitz. Therefore, by Lemma \[HS-1994\_1.3\], we have

\[small\_level\_sets\] Given a domain $D$ and a point $z_0\in \partial D$. Let $R_0$ denote the set of all $r>0$ such that $C_r(z_0)=\{z: |z-z_0|=r\}$ contains uncountably many point components of $\partial D$. If $\partial D$ has $\sigma$-finite linear measure then $R_0$ has zero Lebesgue measure.

A combination of Theorem \[topological-cct\] with the following result will lead us to Theorem \[arsove\_sigma\].

\[oscillation2\] Given a conformal homeomorphism $\varphi: D\rightarrow \Omega$ of a circle domain $D$, where $\partial D$ has $\sigma$-finite linear measure and $\partial\Omega$ has countably many non-degenerate components $\{P_n\}$. Let $Q_n$ be the component of $\partial D$ with $\varphi^B(Q_n)=P_n$ for all $n\ge1$. If there exists an open set $U_n\supset P_n$ satisfying $\displaystyle\sum_{P_k\subset U_n}{\rm diam}(P_k)<\infty$ we have $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial Q_n$.

In Theorem \[oscillation2\], we do not require that $\partial\Omega$ be a Peano compactum. The only assumptions are about the linear measure of $\partial D$ and about the diameters of $P_n$. Therefore, the result we obtain here is just the oscillation convergence $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial Q$. Again, we say nothing about the cluster sets $C(\varphi,z_0)$ for $z_0\in\partial Q$.

For any $r>0$ and $z_0\in Q$, let $C_r(z_0)=\{|z-z_0|=r\}$. By Lemma \[small\_level\_sets\], the boundary components of $D$ that intersect $C_r(z_0)$ forms a countable set for all $r$ except those lying in a set $R_0$ of zero Lebesgue measure.

Let $\Lambda_r(z_0)$ be the [**arc length**]{} of $\varphi(C_r(z_0)\cap D)$. After a slight modification of the proof for Wolff’s Lemma in [@Pom92 p.20, Proposition 2.2], we can show that $$\displaystyle\inf\limits_{\rho<r<\sqrt{\rho},r\notin R_0}\Lambda_r(z_0)\le\frac{2\pi R}{\sqrt{\log1/\rho}}$$ holds for $0<\rho<1$. Therefore, $\liminf\limits_{r\rightarrow 0}\Lambda_r(z_0)=0$ and we can choose for any $\epsilon>0$ a decreasing sequence of numbers outside $R_0$, say $r_1>r_2>\cdots>r_m>\cdots$, such that $\lim\limits_{m\rightarrow 0}r_m=0$ and $\Lambda_{r_m}(z_0)<\frac12\epsilon$ for all $m{\geqslant}1$.

The components of $\partial D$ intersecting $C_{r_m}(z_0)\setminus Q$ for any given $r_m$ form a countable set. Thus we denote them as $\{Q_{k_i}, i=1,2,\cdots\}$. We may assume that every $P_{k_i}=\varphi^B( Q_{k_i})$ lies in the open neighborhood $U_n$ of $P_n=\varphi^B(Q_n)$. This is possible by choosing a sufficiently small $r_1$. Moreover, we may rename $k_i$, if necessary, so that we have $k_1<k_2<\cdots$.

Fix a point $w_0\in \Omega$ and then use [**Zoretti Theorem**]{} to find a simple closed curve $\Gamma_i$ for each $P_{k_i}$ such that [**$\Gamma_i$ separates $w_0$ from $P_{k_i}$**]{} and that [**every point of $\Gamma_i$ is at a distance less than $2^{-i}\epsilon$ from some point of $P_{k_i}$**]{}. Clearly, we have $\displaystyle\sum_{i}{\rm diam}(\Gamma_i)<\infty$. For $i\ge1$, let $W_i^*$ denote the component of $\hat{{\mathbb{C}}}\setminus \Gamma_i$ that contains $P_{k_i}$; moreover, let $W_i$ denote the component of $\hat{{\mathbb{C}}}\setminus \varphi^{-1}(\Gamma_i)$ that contains $Q_{k_i}$. Here $Q_{k_i}$ is the boundary component of $D$ with $\varphi^B\left(Q_{k_i}\right)=P_{k_i}$.

Then, fixing an integer $N\ge1$ with $\displaystyle \sum_{i=N+1}^\infty{\rm diam}(\Gamma_i)<\frac12\epsilon$, we continue to choose a positive number $r\in
\{r_m:m\ge1\}$ that is small enough so that $C_{r}(z_0)\setminus Q$ intersects none of the boundary components $Q_{k_1},\ldots, Q_{k_N}$ of $D$. Moreover, if we let $F_r^*$ be defined as in Lemma \[transboundary\_bridge\], then $F_r^*$ is a countable set and hence we can find a countable open cover $\{V_k^*\}$ of $F_r^*$ such that $\sum_k{\rm diam}(V_k^*)<\frac{\epsilon}{4}$. Consequently, we can follow a similar but simpler argument, as used in Lemma \[estimate\_of\_oscillation\], and verify that for the above $r$, the inequality $|\varphi(z_1)-\varphi(z_2)|<\epsilon$ holds for any fixed points $z_1\ne z_2$ lying on $C_{r}(z_0)\cap D$. This shall complete our proof.

The above proof also works, even if the circle domain $D$ in Theorem \[oscillation2\] is changed into a generalized Jordan domain. Combining this observation with Theorem \[topological-cct\], we actually have the following.

\[arsove\_sigma-new\] Let $\Omega_1$ be a generalized Jordan domain. Let $\varphi: \Omega_1\rightarrow \Omega_2$ be a conformal homeomorphism, where the boundary $\partial\Omega_2$ has at most countably many non-degenerate components $\{P_n\}$ with $\displaystyle\sum_n{\rm diam}(P_n)<\infty$ while all the point components of $\partial\Omega_1$ form a set of $\sigma$-finite linear measure. Then $\varphi$ extends continuously to the closure $\overline{\Omega_1}$ if and only if $\partial\Omega_2$ is a Peano compactum.

To Generalize Osgood-Taylor-Carathéodory Theorem {#final}
================================================

This section addresses on a new generalization of the OTC Theorem, as given in Theorem \[OTC-b\].

We firstly recall some earlier results of a similar nature, which focus on domains that are not far from a circle domain in their metric structure. Then, we give a proof for Theorem \[OTC-b\]. Let us start from four earlier works of a very similar nature. The first comes from an extension theorem by He and Schramm.

Let $\Omega,\Omega^*$ be open connected sets in the Riemann sphere and let $f: \Omega\rightarrow\Omega^*$ be a conformal homeomorphism between them. Let $W$ be an open subset of $B(\Omega)$, which is at most countable. Suppose that the boundary components of $\Omega$ corresponding to elements of $W$ are all circles and points and that the corresponding (under $f$) boundary components of $\Omega^*$ are also circles and points. Then $f$ extends continuously to the boundary components in $W$ and extends to be a homeomorphism between $\bigcup\{K: K\in W\}\cup\Omega$ and $\bigcup\{K^*: K^*\in f^B(W)\}\cup\Omega^*$.

In the above theorem $B(\Omega)$ denotes the space of boundary components of $\Omega$. As a direct corollary we can obtain the following generalization of OTC Theorem.

\[OTC-countable\] Every conformal homeomorphism $\varphi: D\rightarrow\Omega$ of a countably connected circle domain $D$ onto a circle domain $\Omega$ extends to be a homeomorphism between $\overline{D}$ and $\overline{\Omega}$.

In the second, the circle domain $D$ is just required to have a boundary with $\sigma$-finite linear measure. Therefore, it will include as a special case the above Theorem \[OTC-countable\].

\[OTC-1\] Let $D$ be a circle domain in $\hat{{\mathbb{C}}}$ whose boundary has $\sigma$-finite linear measure. Let $\Omega$ be another circle domain and let $\varphi: D\rightarrow\Omega$ be a conformal homeomorphism. Then $\varphi$ extends to be a homeomorphism $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$.

In the third one, the circle domain $D$ is assumed to satisfy the so-called quasihyperbolic condition while $\Omega$ is only required to be a domain whose complement consists of points and a family of uniformly fat closed Jordan domains. Such a domain is just a cofat generalized Jordan domain.

\[OTC-2\] Let $D$ be a circle domain with $\infty\in D$ and let $h$ be conformal map from $D$ onto another domain $\Omega$ with $\infty=h(\infty)\in\Omega$. Suppose that $D$ satisfies the quasihyperbolic condition and that the complementary components of $\Omega$ are uniformly fat closed Jordan domains and points. Then $h$ extends to be a homeomorphism from $\overline{D}$ onto $\overline{\Omega}$.

The last one may be inferred from [@Schramm95 Theorem 6.2], in which $D$ and $\Omega$ are both allowed to be generalized Jordan domains that are cofat.

\[OTC-3\] Let $\varphi: D\rightarrow \Omega$ be a conformal homeomorphism between generalized Jordan domains that are countably connected and cofat. Suppose that for any component $Q$ of $\partial D$ the corresponding component $P=\varphi^B(Q)$ of $\partial\Omega$ is a singleton if and only if $Q$ is a singleton. Then every conformal homeomorphism $\varphi: D\rightarrow\Omega$ of $D$ onto $\Omega$ extends to be a homeomorphism between $\overline{D}$ and $\overline{\Omega}$.

Theorem \[OTC-b\] is comparable with Theorem \[OTC-3\]. There are two major differences. Firstly, we do not require the domains $D,\Omega$ to be countably connected. Secondly, the property of being cofat is replaced by two properties: (1) for one of them the point boundary components form a set of $\sigma$-finite linear measure and (2) for both of them the diameters of the non-degenerate boundary components have a finite sum. Therefore, Theorem \[OTC-b\] is an OTC Theorem for generalized Jordan domains that [**may not be cofat**]{}. Its proof is given as below.

If the point components of $\partial\Omega$ form a set of $\sigma$-finite linear measure we apply Theorem \[arsove-new\] to the map $\varphi: D\rightarrow\Omega$ and obtain a well-defined continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$. Then, applying Theorem \[arsove\_sigma-new\], we see that the inverse map $\psi=\varphi^{-1}: \Omega\rightarrow D$ also extends to be acontinuous map $\overline{\psi}: \overline{\Omega}\rightarrow\overline{D}$. Consequently, we can check that $\overline{\varphi}\circ \overline{\psi}=id_{\overline{\Omega}}$ and $\overline{\psi}\circ \overline{\varphi}=id_{\overline{D}}$. This indicates that $\overline{\varphi}$ and $\overline{\psi}$ are both injective.

If the point components of $\partial D$ form a set of $\sigma$-finite linear measure we apply Theorem \[arsove\_sigma-new\] to the map $\varphi: D\rightarrow\Omega$ and obtain a well-defined continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$. Then, applying Theorem \[arsove-new\] to the inverse map $\psi=\varphi^{-1}: \Omega\rightarrow D$, we obtain another continuous map $\overline{\psi}: \overline{\Omega}\rightarrow\overline{D}$ that extends $\psi$. Similarly, we can infer that $\overline{\varphi}$ and $\overline{\psi}$ are both injective.

[**Acknowledgement**]{}. The authors are grateful to Christopher Bishop at SUNY, to Malik Younsi at University of Hawaii Hanoa, and to Xiaoguang Wang at Zhejiang University for private communications that are of great help, including important references and valuable ideas with concrete mathematical details.

[10]{}

Maynard G. Arsove. Intrinsic characterization of regions bounded by closed curves. , 34:425–429, 1967.

Maynard G. Arsove. The [O]{}sgood-[T]{}aylor-[C]{}arathéodory theorem. , 19:38–44, 1968.

Anders Björn, Jana Björn, and Nageswari Shanmugalingam. The [M]{}azurkiewicz distance and sets that are finitely connected at the boundary. , 26(2):873–897, 2016.

Alexander M. Blokh, Clinton P. Curry, and Lex G. Oversteegen. Locally connected models for [J]{}ulia sets. , 226(2):1621–1661, 2011.

Alexander M. Blokh, Clinton P. Curry, and Lex G. Oversteegen. Finitely [S]{}uslinian models for planar compacta with applications to [J]{}ulia sets. , 141(4):1437–1449, 2013.

Constantin Carath[é]{}odory. Über die [B]{}egrenzung einfach zusammenhängender [G]{}ebiete. , 73(3):323–370, 1913.

Constantin Carath[é]{}odory. Über die gegenseitige [B]{}eziehung der [R]{}änder bei der konformen [A]{}bbildung des [I]{}nneren einer [J]{}ordanschen [K]{}urve auf einen [K]{}reis. , 73(2):305–320, 1913.

E. F. Collingwood and A. J. Lohwater. . Cambridge Tracts in Mathematics and Mathematical Physics, No. 56. Cambridge University Press, Cambridge, 1966.

Clinton P. Curry. Irreducible [J]{}ulia sets of rational functions. , 16(5-6):443–450, 2010.

Adrien Douady. Descriptions of compact sets in [${\bf C}$]{}. In [*Topological methods in modern mathematics ([S]{}tony [B]{}rook, [NY]{}, 1991)*]{}, pages 429–465. Publish or Perish, Houston, TX, 1993.

R. W. FitzGerald and P. M. Swingle. Core decomposition of continua. , 61:33–50, 1967.

Zheng-Xu He and Oded Schramm. Fixed points, [K]{}oebe uniformization and circle packings. , 137(2):369–406, 1993.

Zheng-Xu He and Oded Schramm. Rigidity of circle domains whose boundary has [$\sigma$]{}-finite linear measure. , 115(2):297–310, 1994.

Zheng-Xu He and Oded Schramm. The inverse [R]{}iemann mapping theorem for relative circle domains. , 171(1):157–165, 1995.

Zheng-Xu He and Oded Schramm. Koebe uniformization for “almost circle domains”. , 117(3):653–667, 1995.

David A. Herron. Riemann maps and diameter distance. , 119(2):140–147, 2012.

Jan Kiwi. eal laminations and the topological dynamics of complex polynomials. , 184(2):207–267, 2004.

Paul Koebe. Über die [U]{}niformisierung der algebraischen [K]{}urven. [I]{}. , 67(2):145–224, 1909.

Paul Koebe. Abhandlungen zur [T]{}heorie der konformen [A]{}bbildung. , 2(1-2):198–236, 1918.

K. Kuratowski. . New edition, revised and augmented. Translated from the French by A. Kirkor. Academic Press, New York, 1968.

Benoît Loridant, Jun Luo, and Yi Yang. A core decomposition of compact sets in the plane. , 343:219–244, 2019.

Jun Luo, Yi Yang, and Xiao-Ting Yao. The peano model for rational julia sets. , 2019.

Edwin E. Moise. . Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, Vol. 47.

R. L. Moore. Concerning upper semi-continuous collections of continua. , 27(4):416–428, 1925.

Dimitrios Ntalampekos and Malik Younsi. Rigidity theorems for circle domains. , online, 2019.

William F. Osgood and Edson H. Taylor. Conformal transformations on the boundaries of their regions of definitions. , 14(2):277–298, 1913.

Christian Pommerenke. , volume 299 of [ *Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\]*]{}. Springer-Verlag, Berlin, 1992.

Oded Schramm. Transboundary extremal length. , 66:307–329, 1995.

Kurt Strebel. Über das [K]{}reisnormierungsproblem der konformen [A]{}bbildung. , 1951(101):22, 1951.

William P. Thurston. On the geometry and dynamics of iterated rational maps. In [*Complex dynamics*]{}, pages 3–137. A K Peters, Wellesley, MA, 2009. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher.

Gordon Thomas Whyburn. . American Mathematical Society Colloquium Publications, v. 28. American Mathematical Society, New York, 1942.

Gordon Thomas Whyburn. . Second, revised edition. Princeton Mathematical Series, No. 23. Princeton University Press, Princeton, N.J., 1964.

[^1]: Supported by Chinese National Natural Science Foundation Projects \# 11871483 and 11771391.
