
The \h{Cantor $1/5$-discontinuum} $C_5$ is created using the
procedure of deleting the second and the fourth segments
from five equal segments (instead of the middle third for
the \g{Cantor Ternary Set}). $C_5$ is the set of the reals in
the unit interval which can be represented without $1$-s and
$3$-s in the form
$$
\sum _{n=1} ^\infty \frac{a_n}{5^n} \quad .
$$

The \h{Double Buckethandle} continuum is the union of:

(i) the half circles in the lower plane going
through points of $C_5 \cap \{x : 2/5^{n+1}
\le x \le 1/5^n\}$ with the centre $7/10\cdot 5^n$,


(ii) the half circles in the upper plane going
through points of $C_5 \cap \{x : 2/5^{n+1}
\le 1-x \le 1/5^n\}$ with the centre $1-7/10\cdot
5^n$.


\smallskip

We can find more in \cite[p.205 - p.206]{kuratowski1968}.
