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Cantor organ and accordion

The 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}}$), k\in
\mathbb{N}$ [Kuratowski 1968, p. 191]. See Figure A.

Figure 4.1.2: ( A ) Cantor organ
A.gif

  1. X$ is an arc-like continuum which is irreducible between points (0,x)$ and (1,y)$, where 0\le x,y\le 1$, and has exactly four end points.
  2. It has uncountably many arc components.

A variation of the Cantor organ is the Cantor accordion Y$ which is defined as the monotone image of X$ under a map that shrinks horizontal bars J\times \{0\}$ and J'\times \{1\}$ to points [Kuratowski 1968, p. 191]. See Figure B.

Figure 4.1.2: ( B ) Cantor accordion
B.gif

Besides the above properties,

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next up previous contents index
Next: Arc-like (chainable) continua Up: Elementary examples Previous: Sin curve
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30