Cantor organ and accordion

The Cantor organ is the union of the product of the Cantor ternary set and the unit interval and all segments of the form or , where (, resp.) is the closure of a component of of length ( ), [Kuratowski 1968, p. 191]. See Figure A.

Figure 4.1.2: ( A ) Cantor organ

1. is an arc-like continuum which is irreducible between points and , where , and has exactly four end points.
2. It has uncountably many arc components.

A variation of the Cantor organ is the Cantor accordion which is defined as the monotone image of under a map that shrinks horizontal bars and to points [Kuratowski 1968, p. 191]. See Figure B.

Figure 4.1.2: ( B ) Cantor accordion

Besides the above properties,

• has an upper semi-continuous monotone decomposition into arcs with the quotient space an arc.

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