\h{Semicircle Continuum} is a nice \g{continuum}.


Let $A = \{(x,0)\in \Bbb R^2: 0 \le x \le 1\}$;

for each $n=1,2,\cdots$ and each $k=1,\cdots, 2^{n-1}$, let

$$
B_{n,k} = \left \{(x,y)\in \Bbb R^2:
\left (x-\frac{2k-1}{2^n}\right )^2 + y^2
=4^{-n} \mbox{and} y\ge 0\right \};
$$

for each $n=0,1,\cdots$ and each $k=1,\cdots, 3^{n}$, let

$$
C_{n,k} = \left \{(x,y)\in \Bbb R^2:
\left (x-\frac{2k-1}{2\cdot 3^n}\right )^2 + y^2
=\frac {1}{4}\cdot 9^{-n} \mbox{and} y\le 0\right \};
$$

let

$$
X = A \cup \left [\bigcup_{n=1}^\infty \left (
\bigcup_{k=1}^{2^{n-1}} B_{n,k}\right )\right ]
\cup
\left [\bigcup_{n=0}^\infty \left ( \bigcup_{k=1}^{3^{n}} C_{n,k}\right
)\right ]
$$

\medskip



1) $X$ is an example of an \g{hlc} \g{continuum} which is
not \g{regular}.

2) $X$ is the union of two \g{regular} \g{continua}.

See Nadler 10.59, p.193.
