Some theorems on decomposable continua

In this paper, all spaces are assumed to be metrizable. A $map$ is a continuous single-valued function. A map $f:X\to Y$ is said to be $monotone$ if for each $y\in Y$, $f^{-1}(y)$ is connected. If $X$ is a space and $A$ is a subset of $X$, then we denote the closure of $A$ in $X$ by ${\rm Cl}_{X}A$. By a $continuum$ we mean a compact connected metric space. An $arc$ is a space which is homeomorphic to the closed interval $[0,1]$. If $A$ is an arc, then we denote the set of end points of $A$ by $E(A)$. A continuum is said to be $decomposable$ if it is the union of two proper subcontinua. If a continuum $X$ is not decomposable, then $X$ is said to be $indecomposable$.

Let $X$ be a continuum. Then, $2^{X}$ denotes the space of all closed subsets of $X$ with the topology generated by the Hausdorff metric. Also, $C(X)$ denotes the subspace of $2^{X}$ consisting of all nonempty subcontinua of $X$. A $Whitney$ $map$ is a map $\mu:C(X)\to[0,\mu(X))$ satisfying $\mu(\{x\})=0$ for each $x\in X$, and $\mu(A)<\mu(B)$ whenever $A,B\in C(X)$ and $A\subsetneq B$. It is well-known that for each Whitney map $\mu:C(X)\to[0,\mu(X)]$ and for each $t\in[0,\mu(X)]$, $\mu^{-1}(t)$ is a continuum ([4, Theorem 19.9]).

A topological property $P$ is called a Whitney property if a continuum $X$ has property $P$, so does $\mu^{-1}(t)$ for any Whitney map $\mu$ for $C(X)$ and for any $t\in[0,\mu(X))$. Also, a topological property $P$ is called a Whitney reversible property provided that whenever $X$ is a continuum such that $\mu^{-1}(t)$ has property $P$ for each Whitney map $\mu$ for $C(X)$ and for each $t\in(0,\mu(X))$, then $X$ has property $P$. For information about Whitney properties and Whitney reversible properties, see [4, Chapter 8].

In this paper, we show some theorems on decomposable continua. In particular, we deal with topics on Wilder continua, $D$-continua, $D^{*}$-continua and $D^{**}$-continua (definitions of these continua are given in the following section). Studies on these continua are seen in [1], [2], [10], [11], [12], [13], [14] and [18].

The paper is divided as follows:

In Section 2, we introduce some basic definitions and terminology that are used throughout this paper.

In Section 3, we show that the properties of being a Wilder continuum is not a Whitney reversible property. This result gives a partial answer to [14, Qusetion 5.2]. Note that the properties of being a Wilder continuum, being a $D$-continuum, being a $D^{*}$-continuum and being a $D^{**}$-continuum are all Whitney properties (see [14]).

In Section 4, we deal with topics on inverse limits with set-valued bonding functions. The notion of inverse limits with set-valued bonding functions was introduced by Ingram and Mahavier ([5]). In Section 4, we prove that inverse limits of $D^{**}$-continua with surjective monotone upper semi-continuous bonding functions are $D^{**}$. This result is in contrast to [17, Examples 5.1 and 5.2], which state that there exists an inverse sequence $\{X_{i},f_{i}\}_{i=1}^{\infty}$ of $D$-continua (resp. $D^{*}$-continua) with surjective monotone upper semi-continuous bonding functions such that $\underleftarrow{{\rm lim}}\{X_{i},f_{i}\}_{i=1}^{\infty}$ is not a $D$-continuum (resp. a $D^{*}$-continuum).

In Section 5, we show that there exists a $D^{**}$-continuum which contains neither Wilder continua nor $D^{*}$-continua. Also, we show the existence of a Wilder continuum containing no $D^{*}$-continua and a $D^{*}$-continuum containing no Wilder continua.

In this section, we introduce some basic definitions and terminology.

Wilder introduced the notion of Wilder continua in [18].¹¹1Wilder named the continua as $C$-continua. However, we use the term Wilder continua following the notion proposed in [8].

Let $X$ be a continuum. If $X$ is not Wilder, then there exist three distinct points $x,y,z\in X$ such that for each subcontinuum $C$ of $X$ with $x\in C$ and $\{y,z\}\cap C\neq\emptyset$, $\{y,z\}\subseteq C$. In this case, we say that $y$ and $z$ are $points$ $of$ $non$-$Wilderness$ $of$ $X$.

Note that a continuum $X$ is a $D^{**}$-continuum if and only if for each $a\in X$ and for each nondegenerate subcontinuum $B$ of $X$ with $a\notin B$, there exists a subcontinuum $C$ of $X$ such that $a\in C$, $B\cap C\neq\emptyset$ and $B\setminus C\neq\emptyset$.

It is clear that every $D^{*}$-continuum is a $D^{**}$-continuum and every $D^{**}$-continuum is a $D$-continuum. Note that the $\sin\frac{1}{x}$-continuum is a $D$-continuum which is not a $D^{**}$-continuum.

In [14, Section 2], it is proven that the classes of Wilder continua and $D^{*}$-continua are strictly contained in the class of $D^{**}$-continua. Since every nondegenerate indecomposable continuum has uncountably many composants [16, Theorem 11.15], we can easily see that every nondegenerate $D$-continuum is decomposable. For other relationships between the classes of the above continua and other classes of continua, see [2, Figure 6].

Let $X$ and $Y$ be any compacta. Then, a set-valued function $f:X\to 2^{Y}$ is called an $upper$ $semi$-$continuous$ $function$ provided that for each $x\in X$ and each open subset $V$ of $Y$ containing $f(x)$, there exists an open subset $U$ of $X$ which contains $x$ such that if $z\in U$, then $f(z)\subset V$. The $graph$ of a set-valued function $f:X\to 2^{Y}$ is the set $G(f)=\{(x,y)\in X\times Y~|~y\in f(x)\}$. We say that a set-valued function $f:X\to 2^{Y}$ is $surjective$ if for each $y\in Y$, there exists $x\in X$ such that $y\in f(x)$.

Every map $f:X\to Y$ induces the upper semi-continuous function $\tilde{f}:X\to 2^{Y}$ defined by $\tilde{f}(x)=\{f(x)\}$ for each $x\in X$. Therefore, if necessary, we may regard a map $f:X\to Y$ as the upper semi-continuous function $\tilde{f}:X\to 2^{Y}$.

An $inverse$ $sequence$ is a double sequence $\{X_{i},f_{i}\}_{i=1}^{\infty}$ of compacta $X_{i}$ and upper semi-continuous functions $f_{i}:X_{i+1}\to 2^{X_{i}}$. The spaces $X_{i}$ are called the $factor$ $spaces$ and the upper semi-continuous functions $f_{i}$ the $bonding$ $functions$. We denote the subspace $\{(x_{i})_{i=1}^{\infty}\in\prod_{i=1}^{\infty}X_{i}~|~$for each $i\geq 1,~x_{i}\in f_{i}(x_{i+1})\}$ of $\prod_{i=1}^{\infty}X_{i}$ by $\underleftarrow{{\rm lim}}\{X_{i},f_{i}\}_{i=1}^{\infty}$. $\underleftarrow{{\rm lim}}\{X_{i},f_{i}\}_{i=1}^{\infty}$ is also denoted by $X_{\infty}$. $X_{\infty}$ is called the $inverse$ $limit$ $of$ $\{X_{i},f_{i}\}_{i=1}^{\infty}$.

First, we show the following result which can yield a lot of examples of non-Wilder continua whose all positive Whitney levels are Wilder continua.

By the following example, we see that the property of being a Wilder continuum is not a Whitney reversible property. This result gives a partial answer to [14, Question 5.2].

In this section, we deal with topics on inverse limits with set-valued bonding functions.

Note that if a map $f:X\to Y$ between compacta is monotone, then it satisfies the condition of $f$ in the following definition.

The following results are proven in [17].

The main aim of this section is to prove Theorem 4.8, which is in contrast to the above results. To prove the theorem, we need Theorem 4.6 proven in [17].

Furthermore, we need the following result. The proof is similar to the proof of [2, Theorem 4.3]. Hence, we omit the proof.

The following theorem is the main result in this section.

Let $X$ be a nondegenerate continuum. Then $X$ is said to be arc-like if for each $\varepsilon>0$, there exists a surjective map $f:X\to[0,1]$ such that for each $y\in[0,1]$, ${\rm diam}f^{-1}(y)<\varepsilon$. A continnuum is said to be $hereditarily$ $decomposable$ if each of its all subcontinua is decomposable.

A continuum $X$ is said to be hereditarily arcwise connected if each of its subcontinuum is arcwise connected. Also, if each subcontinuum of $X$ is a Wilder continuum (resp. a $D$-continuum, a $D^{*}$-continuum, a $D^{**}$-continuum), then $X$ is called a hereditarily Wilder continuum (resp. a hereditarily $D$-continuum, a hereditarily $D^{*}$-continuum, a hereditarily $D^{**}$-continuum).

Hence, it is natural to ask whether or not there exists a hereditarily $D^{**}$-continuum containing no $D^{*}$-continua. The answer to this question is negative by the following result.

However, in Theorem 5.4, we show that there exists a $D^{**}$-continuum containing neither Wilder continua nor $D^{*}$-continua.

Let $B$ be the subcontinuum of $\mathbb{R}^{2}$ defined by

(see Figure 3). Let $P$ and $Q$ be the subcontinua of $B$ defined by

Also, let $R^{0}$, $Q^{0}$, $R^{1}$ and $Q^{1}$ be the subspaces of $B$ defined by

Then, $R^{0}$, $Q^{0}$, $R^{1}$ and $Q^{1}$ are homeomorphic to $[0,1)$ respectively. Also, note that ${\rm Cl}_{B}R^{0}$, ${\rm Cl}_{B}Q^{0}$, ${\rm Cl}_{B}R^{1}$ and ${\rm Cl}_{B}Q^{1}$ are homeomorphic to the $\sin\frac{1}{x}$-continuum. Furthermore, let $a$, $b$, $p$, $p^{\prime}$, $q$, $q^{\prime}$, $s$, $s^{\prime}$, $t$, $t^{\prime}$, $u$, $u^{\prime}$, $v$, $v^{\prime}\in B$ be the points as in Figure 3. Finally, let $S(B)=\{a$, $b$, $p$, $p^{\prime}$, $q$, $q^{\prime}$, $s$, $s^{\prime}$, $t$, $t^{\prime}$, $u$, $u^{\prime}$, $v$, $v^{\prime}\}$.

In the following theorem, if $\{X_{i},f_{i}\}_{i=0}^{\infty}$ is an inverse sequence with single valued bonding maps, then we denote the map $f_{i}\circ\cdots\circ f_{j-1}:X_{j}\to{X_{i}}$ by $f_{i,j}$ if $j>i+1$. Also, let $f_{i,i+1}=f_{i}$. Furthermore, for each $n\geq 0$, let $\pi_{n}:X_{\infty}\to X_{n}$ be the projection. Finally, if $f:X\to Y$ is a map and $y\in Y$ is a point such that $f^{-1}(y)$ is a one point set, then we also denote the point in $f^{-1}(y)$ by $f^{-1}(y)$.