{Sin curve}

The
\eS{(topologist's)}{topologist's sin curve}
\e{sin curve}
$S$ is defined by
$$S=\{\,(x,y)\in \mathbb R^2: y=\sin\frac1x, 0<x\le 1\,\}\cup \{\,(0,y):-1\le
y\le 1\,\}.$$
\rostere
\item
$S$ is \g{irreducible} between points $(0,y)$ and
$(1,\sin 1)$, $-1\le y\le 1$. It has exactly three
\gs{end points}{end point} and two arc components.
\item
It is one of the simplest \g{arc-like} continua.
\item
It is a \g{compactification} of a ray
$(0,1]$ with remainder an arc.
\item It has the \g{periodic-recurrent
property} \cite[Corollary 5.10, p. 117]{Charatonik+1997b}.
\item
The only possible \g{confluent} nondegenerate
images of $S$ are an arc and a continuum homeomorphic to $S$
\cite{Nadler1977a}.
\item
The \g{hyperspace} $C(S)$ of all subcontinua of
$S$ is homeomorphic to the \g{cone} over $S$
\cite{Nadler1977b}.
\rosteree

See Figure A.

\FIGURE{A} {sin curve}

There are many variations of the sin curve. Some of them are pictured below.
See Figure B--C.


\FIGURE{B}{union of two sin curves}

\FIGURE{C}{union of two sin curves}
