Buckethandle

The Buckethandle is created from the Cantor Ternary Set C with this procedure: (i) we join any two points a and b in C symmetric with respect to 1/2 with a semicircle in the upper half plane with the centre in (1/2,0), (ii) we join any two points of C in the interval $2/3^n \le x \le 3/3^n$, $n\ge 1$, with a semicircle with the centre in $(5/(2\cdot 3^n),0)$ in the lower half plane. This continuum is often called the Knaster's Buckethandle continuum (see Figure (A)).

This kind of curiosities in continuum theory is connected with common boundary of several plane regions. In 1904 A. Schönflies started to publish a sequence of papers [6] which became an important step in development of continuum theory. Relying heavily on intuition, he claimed that there do not exist three regions in the plane with common boundary. The claim was refused by L. E. J. Brouwer in 1910 [1] who constructed continua which are common boundary of three regions and showed that they are indecomposable, i.e., such continua X that there are no nonempty proper subcontinua A and B of X with $X = A \cup B$. The first example of such a continuum was given by Brouwer in 1910 (see Figure B, where the first several steps of his construction, simplified by Z. Janiszewski (1888-1920) [2, p. 114] are presented). Finally in 1922 B. Knaster (1893-1980) gave a nice description of this continuum (with the first full proof of its indecomposability) in [3, p. 209-210]. Now the continuum is referred to as the simplest indecomposable continuum, or the horse-shoe continuum, or the B-J-K continuum (for Brouwer, Janiszewski and Knaster).

Finally the common boundary problem of plane domains has been solved in 1928 by K. Kuratowski (1896-1980) who proved in [4] and [5] that every plane continuum which is the common boundary of n open domains either is indecomposable or is the union of two indecomposable continua whenever $n \ge 3$, and when n = 2 it either is "monostratic" or has a natural "cyclic structure" in the sense that it is built up from layers naturally ordered in the same way as the individual points of the circle.

We can obtain a homeomorphic copy of the buckethandle continuum using the inverse limit

$$\lim \limits _{\leftarrow} \{X_i,f_i\}_{i=1}^\infty$$

where for each $i=1,2,\cdots$ let X_i=[0,1] and f_i(t)=2t for $0\le t \le 1/2$ and f_i(t)=-2t+2 for $1/2\le t \le 1$.

We can find more in [65, p.22] and [55, p.205]. [1] L. E. J. Brouwer : Zur Analysis Situs, Math. Ann. 68(1910), 422-434. [2] Z. Janiszewski: Sur les continus irréductibles entre deux points, Journal de l'Ecole Polytechnique (2) 16(1912), 79-170. [3] K. Kuratowski: Théorie des continus irréductibles entre deux points I, Fund. Math. 3(1922), 200-231. [4] K. Kuratowski: Sur les coupures irréductibles du plan, Fund. Math. 6(1924), 130-145. [5] K. Kuratowski: Sur la structure des frontières communes à deux regions, Fund. Math. 12(1928), 20-42. [6] A. Schönflies: Beirträge zur Theorie der Punktmengen I, Math. Ann. 58(1904), 195-244; II 59(1904), 129-160; III 62(1906), 286-236.

Figure ( A ) Buckethandle


Figure ( B ) several steps in the construction of the Buckethandle


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