{Cantor organ and accordion}

The
\e{Cantor organ}
$X$ is the union of the product $C\times I$ of the Cantor
ternary set $C$ and the unit interval $I$ and  all segments
of the form  $J\times \{0\}$ or $J'\times \{1\}$, where $J$
($J'$, resp.) is the closure of a component of $I\setminus
C$ of length $1/{3^{2k-1}}$ ($1/{3^{2k}}$), $k\in
\mathbb N$ \cite[p. 191]{Kuratowski1968a}. See Figure A.

\FIGURE{A}{Cantor organ}

\rostere
\item
$X$ is an \g{arc-like} \g{continuum} which is
\g{irreducible} between points $(0,x)$ and $(1,y)$, where
$0\le x,y\le 1$, and  has exactly four
\gs{end points}{end point}.
\item
It has uncountably many arc components.
\rosteree


A variation of the Cantor organ is the
\e{Cantor accordion}
$Y$ which is defined as the \gs{monotone}{monotone map}
image of $X$ under a map that shrinks horizontal bars
$J\times \{0\}$ and $J'\times \{1\}$ to points \cite[p.
191]{Kuratowski1968a}. See Figure B.

\FIGURE{B}{Cantor accordion}
%Figure as in \cite[p. 191]{Kuratowski1968a} (the right one)

Besides the above properties,
\rosteri
\item
$Y$ has an \g{upper semi-continuous}
\g{monotone decomposition} into arcs with the quotient space
an arc.
\rosterii
