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\put(143,66){\makebox(0,0)[tl]{\footnotesize Proceedings of the Ninth Prague Topological Symposium}}
\put(143,50){\makebox(0,0)[tl]{\footnotesize Contributed papers from the symposium held in}}
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\title{Chainable subcontinua}
\author{Edwin Duda}
\address{University of Miami, Department of Mathematics\\
PO Box 249085\\
Coral Gables, FL 33124-4250}
\email{e.duda@math.miami.edu}
\subjclass[2000]{54F20}
\keywords{chainable continuum}
\begin{abstract}
This paper is concerned with conditions under which a metric continuum (a
compact connected metric space) contains a non-degenerate chainable
continuum.
\end{abstract}
\thanks{Edwin Duda,
{\em Chainable subcontinua},
Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001),
pp.~71--73, Topology Atlas, Toronto, 2002; {\tt arXiv:math.GN/0204122}}
\maketitle

This paper is concerned with conditions under which a metric continuum (a
compact connected metric space) contains a non-degenerate chainable 
continuum.

By R.H. Bing's theorem eleven \cite{MR13:265a}
if a metric continuum $X$ contains a non-degenerate
subcontinuum $H$ which is hereditarily decomposable, hereditarily
unicoherent, and atriodic, then $H$ is chainable.

The following papers give examples of continua with the property that each
non-degenerate subcontinuum is not chainable. G.T. Whyburn 
\cite{whyburn}.
R.D. Anderson and G. Choquet
\cite{MR21:3819}.
A. Lelek 
\cite{MR26:742}
gives an example of a planar weakly chainable continuum each
non-degenerate subcontinuum of which separates the plane
and thus contains no non-degenerate chainable subcontinuum. 
W.T. Ingram
\cite{MR82k:54056}
gives an example of an hereditarily
indecomposable tree-like continuum such that each non-degenerate
subcontinuum has positive span and hence is not chainable.

C.E. Burgess in 
\cite{MR23:A3551}
shows if a continuum $M$ is almost chainable and $K$ is a proper
subcontinuum of $M$ which contains an endpoint $p$ of $M$, then $K$ is
linearly chainable with $p$ as an end point. 
A continuum $M$ is almost chainable if, for every positive number 
$\varepsilon $, there exists an $\varepsilon$-covering $G$ of $M$ and a
linear chain $C(L_1,L_2,\ldots,L_n)$ of elements of $G$ such that no
$L_i$ $(1\leq i<n)$ intersects an element of $G-C$ and every point of $M$
is within a distance $\varepsilon$ of some element of $C$. 
He also shows if $M$ is almost chainable, then $M$ is not a triod and $M$
is unicoherent and irreducible between some two points. 
Examples show $M$ can contain a triod or a non-unicoherent subcontinuum.

If $X$ and $Y$ are metric continua and if $X$ can be $\varepsilon$-mapped
onto $Y$ for all positive $\varepsilon $ and $Y$ has a non-degenerate
chainable continuum then so does $X$. 
This result suggests considering inverse limit spaces. 
At this stage we refer to a result from the paper of
S. Marde\u{s}i\'{c} and J. Segal
\cite{MR28:1592}
Theorem 1, p.\ 148: ``Every $\pi $-like continuum $X$ is the inverse limit
of an inverse sequence $\{P_i;\pi _{ij}\}$ with bonding maps $\pi _{ij}$
onto and with polyhedra $P_i\in \pi$. 
A continuum is $\pi $-like if it can be $\varepsilon $-mapped onto some
polyhedron in $\pi $ for each positive $\varepsilon $. 
E. Duda and P. Krupski 
\cite{MR93a:54031}
showed that a $k$-junctioned metric continuum, $k$ a non-negative integer,
has at most $k$ points such that any continuum which contains none of the
$k$ points is chainable. A metric continuum is said to be $k$-junctioned
if it is the inverse limit of graphs each of which has at most $k$ branch
points, with surjective bonding maps. A continuum is called finitely
junctioned if it is $k$-junctioned for some non negative integer $k$.

Suppose now $X$ is a tree-like continuum. 
Then for each $\varepsilon >0$ $\;X$ can be mapped onto a tree. 
By a result quoted above $X$ is the inverse limit of a sequence of trees
with surjective bonding maps. 
$X={\displaystyle \lim _{\longleftarrow }}\{T_n,f_{nm} \}$.
Let $f_n:X\rightarrow T_n$ be the standard projection map and let
$$P_n=U\{f^{-1}_n(q)|q \; \mbox{is a branch point}\}.$$ 
Since $T_n$ has at most a finite number of branch points (points of order
$\geq 2$) $P_n$ is closed in $X$. 
If the union of the $P_n$ is not dense in $X$ then $X$ contains a
non-degenerate chainable continuum. 
Actually it is sufficient that $\{P_n\}$ have a subsequence whose union is
not dense in $X$.

Lets now consider a non-degenerate metric continuum in $X$ with span equal
to zero. 
The notion of span was defined by A. Lelek
\cite{MR31:4009}.
In the paper
\cite{MR82c:54031}
he showed continua with span zero are atriodic and tree-like.

There is a series of papers by L.G. Oversteegen and E.D. Tymchatyn which
develop properties of spaces with spans equal to zero or sufficient
conditions that a space have a span equal to zero 
\cite{MR84h:54030, MR86a:54042, MR85j:54051, MR85m:54034}.
Also by L.G. Oversteegen 
\cite{MR91g:54049}.

It is interesting to note that a chainable continuum $X$ can be
$\varepsilon $-mapped onto any fixed dendrite. 
Thus for any tree $T$, by the result of Marde\u{s}i\'{c} and Segal quoted
above, $X$ is the inverse of a sequence of $T$'s.

In the paper 
\cite{MR95c:54056}
P. Minc shows an inverse limit of trees with
simplicial bonding maps having surjective span zero is chainable.

%\bibliographystyle{amsplain}
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\begin{thebibliography}{10}

\bibitem{MR21:3819}
R.~D. Anderson and Gustave Choquet, \emph{A plane continuum no two of whose
  nondegenerate subcontinua are homeomorphic: {A}n application of inverse
  limits}, Proc. Amer. Math. Soc. \textbf{10} (1959), 347--353. \MR{21 \#3819}

\bibitem{MR13:265a}
R.~H. Bing, \emph{Snake-like continua}, Duke Math. J. \textbf{18} (1951),
  653--663. \MR{13,265a}

\bibitem{MR23:A3551}
C.~E. Burgess, \emph{Homogeneous continua which are almost chainable}, Canad.
  J. Math. \textbf{13} (1961), 519--528. \MR{23 \#A3551}

\bibitem{MR93a:54031}
Edwin Duda and Pawe{\l} Krupski, \emph{A characterization of finitely
  junctioned continua}, Proc. Amer. Math. Soc. \textbf{116} (1992), no.~3,
  839--841. \MR{93a:54031}

\bibitem{MR82k:54056}
W.~T. Ingram, \emph{Hereditarily indecomposable tree-like continua. {I}{I}},
  Fund. Math. \textbf{111} (1981), no.~2, 95--106. \MR{82k:54056}

\bibitem{MR26:742}
A.~Lelek, \emph{On weakly chainable continua}, Fund. Math. \textbf{51}
  (1962/1963), 271--282. \MR{26 \#742}

\bibitem{MR31:4009}
\bysame, \emph{Disjoint mappings and the span of spaces}, Fund. Math.
  \textbf{55} (1964), 199--214. \MR{31 \#4009}

\bibitem{MR82c:54031}
\bysame, \emph{The span of mappings and spaces}, Topology Proc. \textbf{4}
  (1979), no.~2, 631--633. \MR{82c:54031}

\bibitem{MR28:1592}
Sibe Marde{\v{s}}i{\'c} and Jack Segal, \emph{$\varepsilon $-mappings onto
  polyhedra}, Trans. Amer. Math. Soc. \textbf{109} (1963), 146--164. \MR{28
  \#1592}

\bibitem{MR95c:54056}
Piotr Minc, \emph{On simplicial maps and chainable continua}, Topology Appl.
  \textbf{57} (1994), no.~1, 1--21. \MR{95c:54056}

\bibitem{MR91g:54049}
Lex~G. Oversteegen, \emph{On span and chainability of continua}, Houston J.
  Math. \textbf{15} (1989), no.~4, 573--593. \MR{91g:54049}

\bibitem{MR84h:54030}
Lex~G. Oversteegen and E.~D. Tymchatyn, \emph{Plane strips and the span of
  continua. {I}}, Houston J. Math. \textbf{8} (1982), no.~1, 129--142.
  \MR{84h:54030}

\bibitem{MR85j:54051}
\bysame, \emph{On the span of weakly-chainable continua}, Fund. Math.
  \textbf{119} (1983), no.~2, 151--156. \MR{85j:54051}

\bibitem{MR85m:54034}
\bysame, \emph{On span and weakly chainable continua}, Fund. Math. \textbf{122}
  (1984), no.~2, 159--174. \MR{85m:54034}

\bibitem{MR86a:54042}
\bysame, \emph{Plane strips and the span of continua. {I}{I}}, Houston J. Math.
  \textbf{10} (1984), no.~2, 255--266. \MR{86a:54042}

\bibitem{whyburn}
G.~T. Whyburn, \emph{A continuum every subcontinuum of which separate the
  plane}, Amer. J. Math. \textbf{52} (1930), 319--330.

\end{thebibliography}

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