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Sin curve

The (topologist's) sin curve S$ is defined by

\displaystyle S=\{\,(x,y)\in \mathbb{R}2: y=\sin\frac1x, 0<x\le 1\,\}\cup \{\,(0,y):-1\le
y\le 1\,\}.$

  1. S$ is irreducible between points (0,y)$ and (1,\sin 1)$, -1\le y\le 1$. It has exactly three end points and two arc components.
  2. It is one of the simplest arc-like continua.
  3. It is a compactification of a ray (0,1]$ with remainder an arc.
  4. It has the periodic-recurrent property [Charatonik et al. 1997b, Corollary 5.10, p. 117].
  5. The only possible confluent nondegenerate images of S$ are an arc and a continuum homeomorphic to S$ [Nadler 1977a].
  6. The hyperspace C(S)$ of all subcontinua of S$ is homeomorphic to the cone over S$ [Nadler 1977b].
See Figure A.

Figure 4.1.1: ( A ) sin curve
A.gif

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

Figure 4.1.1: ( B ) union of two sin curves
B.gif

Figure 4.1.1: ( C ) union of two sin curves
C.gif

Here you can find source files of this example.

Here you can check the table of properties of individual continua.

Here you can read Notes or write to Notes ies of individual continua.
next up previous contents index
Next: Cantor organ and accordion Up: Elementary examples Previous: Elementary examples
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30