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.

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].

Buckethandle - fig. a


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