
The \h{Buckethandle} is created from the
\g{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
\h{Knaster's Buckethandle} continuum.

\smallskip

We can obtain a homeomorphic copy of the buckethandle
continuum using the \g{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 $1/2\le t \le 1$.

\smallskip

We can find more in \cite[p.22]{nadler1992}  and
\cite[p.205]{kuratowski1968}.
