<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\h{Boxes Continuum} is a nice \g{continuum}.


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

for each $n=0,1,2,\cdots$ and each $m=0,\cdots, 2^{n+1}$,
let

$$
B_{n,m} = \left \{(m\cdot 2^{-n-1},y)\in \Bbb R^2: 0\le y
\le2^{-n}\right \};
$$


finally, let

$$
X = \left [\bigcup_{n=1}^\infty A_n\right ]
\cup
\left [\bigcup_{n=0}^\infty \left ( \bigcup_{m=0}^{2^{n+1}} B_{n,m}\right
)\right ].
$$

\medskip



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

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

See Nadler 10.38, p.186.
</pre></body></html>